## Another Example on Calculating Covariance

In a previous post called An Example on Calculating Covariance, we calculated the covariance and correlation coefficient of a discrete joint distribution where the conditional mean $E(Y \lvert X=x)$ is a linear function of $x$. In this post, we give examples in the continuous case. Problem A is worked out and Problem B is left as exercise.

The examples presented here are also found in the post called Another Example of a Joint Distribution. Some of the needed calculations are found in this previous post.

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Problem A
Let $X$ be a random variable with the density function $f_X(x)=\alpha^2 \ x \ e^{-\alpha x}$ where $x>0$. For each realized value $X=x$, the conditional variable $Y \lvert X=x$ is uniformly distributed over the interval $(0,x)$, denoted symbolically by $Y \lvert X=x \sim U(0,x)$. Obtain solutions for the following:

1. Calculate the density function, the mean and the variance for the conditional variable $Y \lvert X=x$.
2. Calculate the density function, the mean and the variance for the conditional variable $X \lvert Y=y$.
3. Use the fact that the conditional mean $E(Y \lvert X=x)$ is a linear function of $x$ to calculate the covariance $Cov(X,Y)$ and the correlation coefficient $\rho$.

Problem B
Let $X$ be a random variable with the density function $f_X(x)=4 \ x^3$ where $0. For each realized value $X=x$, the conditional variable $Y \lvert X=x$ is uniformly distributed over the interval $(0,x)$, denoted symbolically by $Y \lvert X=x \sim U(0,x)$. Obtain solutions for the following:

1. Calculate the density function, the mean and the variance for the conditional variable $Y \lvert X=x$.
2. Calculate the density function, the mean and the variance for the conditional variable $X \lvert Y=y$.
3. Use the fact that the conditional mean $E(Y \lvert X=x)$ is a linear function of $x$ to calculate the covariance $Cov(X,Y)$ and the correlation coefficient $\rho$.

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Background Results

Here’s the idea behind the calculation of correlation coefficient in this post. Suppose $X$ and $Y$ are jointly distributed. When the conditional mean $E(Y \lvert X=x)$ is a linear function of $x$, that is, $E(Y \lvert X=x)=a+bx$ for some constants $a$ and $b$, it can be written as the following:

$\displaystyle E(Y \lvert X=x)=\mu_Y + \rho \ \frac{\sigma_Y}{\sigma_X} \ (x - \mu_X)$

Here, $\mu_X=E(X)$ and $\mu_Y=E(Y)$. The notations $\sigma_X$ and $\sigma_Y$ refer to the standard deviation of $X$ and $Y$, respectively. Of course, $\rho$ refers to the correlation coefficient in the joint distribution of $X$ and $Y$ and is defined by:

$\displaystyle \rho=\frac{Cov(X,Y)}{\sigma_X \ \sigma_Y}$

where $Cov(X,Y)$ is the covariance of $X$ and $Y$ and is defined by

$Cov(X,Y)=E[(X-\mu_X) \ (Y-\mu_Y)]$

or equivalently by $Cov(X,Y)=E(X,Y)-\mu_X \mu_Y$.

Just to make it clear, in the joint distribution of $X$ and $Y$, if the conditional mean $E(X \lvert Y=y)$ is a linear function of $y$, then we have:

$\displaystyle E(X \lvert Y=y)=\mu_X + \rho \ \frac{\sigma_X}{\sigma_Y} \ (y - \mu_Y)$

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Discussion of Problem A

Problem A-1

Since for each $x$, $Y \lvert X=x$ has the uniform distribution $U(0,x)$, we have the following:

$\displaystyle f_{Y \lvert X=x}=\frac{1}{x}$ for $x>0$

$\displaystyle E(Y \lvert X=x)=\frac{x}{2}$

$\displaystyle Var(Y \lvert X=x)=\frac{x^2}{12}$

Problem A-2

In a previous post called Another Example of a Joint Distribution, the joint density function of $X$ and $Y$ is calculated to be: $f_{X,Y}(x,y)=\alpha^2 \ e^{-\alpha x}$. In the same post, the marginal density of $Y$ is calculated to be: $f_Y(y)=\alpha e^{-\alpha y}$ (exponentially distributed). Thus we have:

\displaystyle \begin{aligned} f_{X \lvert Y=y}(x \lvert y)&=\frac{f_{X,Y}(x,y)}{f_Y(y)} \\&=\frac{\alpha^2 \ e^{-\alpha x}}{\alpha \ e^{-\alpha \ y}} \\&=\alpha \ e^{-\alpha \ (x-y)} \text{ where } y

Thus the conditional variable $X \lvert Y=y$ has an exponential distribution that is shifted to the right by the amount $y$. Thus we have:

$\displaystyle E(X \lvert Y=y)=\frac{1}{\alpha}+y$

$\displaystyle Var(Y \lvert X=x)=\frac{1}{\alpha^2}$

Problem A-3

To compute the covariance $Cov(X,Y)$, one approach is to use the definition indicated above (to see this calculation, see Another Example of a Joint Distribution). Here we use the idea that the conditional mean $\displaystyle E(Y \lvert X=x)$ is linear in $x$. From the previous post Another Example of a Joint Distribution, we have:

$\displaystyle \sigma_X=\frac{\sqrt{2}}{\alpha}$

$\displaystyle \sigma_Y=\frac{1}{\alpha}$

Plugging in $\sigma_X$ and $\sigma_Y$, we have the following calculation:

$\displaystyle \rho \ \frac{\sigma_Y}{\sigma_X}=\frac{1}{2}$

$\displaystyle \rho = \frac{\sigma_X}{\sigma_Y} \times \frac{1}{2}=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}=0.7071$

$\displaystyle Cov(X,Y)=\rho \ \sigma_X \ \sigma_Y=\frac{1}{\alpha^2}$

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Problem B-1

$\displaystyle E(Y \lvert X=x)=\frac{x}{2}$

$\displaystyle Var(Y \lvert X=x)=\frac{x^2}{12}$

Problem B-2

$\displaystyle f_{X \lvert Y=y}(x \lvert y)=\frac{4 \ x^2}{1-y^3}$ where $0 and $y

Problem B-3

$\displaystyle \rho=\frac{\sqrt{3}}{2 \ \sqrt{7}}=0.3273268$

$\displaystyle Cov(X,Y)=\frac{1}{75}$

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$\copyright \ 2013$

## Another Example of a Joint Distribution

In an earlier post called An Example of a Joint Distribution, we worked a problem involving a joint distribution that is constructed from taking product of a conditional distribution and a marginial distribution (both discrete distributions). In this post, we work on similar problems for the continuous case. We work problem A. Problem B is left as exercises.

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Problem A
Let $X$ be a random variable with the density function $f_X(x)=\alpha^2 \ x \ e^{-\alpha x}$ where $x>0$. For each realized value $X=x$, the conditional variable $Y \lvert X=x$ is uniformly distributed over the interval $(0,x)$, denoted symbolically by $Y \lvert X=x \sim U(0,x)$. Obtain solutions for the following:

1. Discuss the joint density function for $X$ and $Y$.
2. Calculate the marginal distribution of $X$, in particular the mean and variance.
3. Calculate the marginal distribution of $Y$, in particular, the density function, mean and variance.
4. Use the joint density in part A-1 to calculate the covariance $Cov(X,Y)$ and the correlation coefficient $\rho$.

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Problem B
Let $X$ be a random variable with the density function $f_X(x)=4 \ x^3$ where $0. For each realized value $X=x$, the conditional variable $Y \lvert X=x$ is uniformly distributed over the interval $(0,x)$, denoted symbolically by $Y \lvert X=x \sim U(0,x)$. Obtain solutions for the following:

1. Discuss the joint density function for $X$ and $Y$.
2. Calculate the marginal distribution of $X$, in particular the mean and variance.
3. Calculate the marginal distribution of $Y$, in particular, the density function, mean and variance.
4. Use the joint density in part B-1 to calculate the covariance $Cov(X,Y)$ and the correlation coefficient $\rho$.

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Discussion of Problem A

Problem A-1

The support of the joint density function $f_{X,Y}(x,y)$ is the unbounded lower triangle in the xy-plane (see the shaded region in green in the figure below).

Figure 1

The unbounded green region consists of vertical lines: for each $x>0$, $y$ ranges from $0$ to $x$ (the red vertical line in the figure below is one such line).

Figure 2

For each point $(x,y)$ in each vertical line, we assign a density value $f_{X,Y}(x,y)$ which is a positive number. Taken together these density values sum to 1.0 and describe the behavior of the variables $X$ and $Y$ across the green region. If a realized value of $X$ is $x$, then the conditional density function of $Y \lvert X=x$ is:

$\displaystyle f_{Y \lvert X=x}(y \lvert x)=\frac{f_{X,Y}(x,y)}{f_X(x)}$

Thus we have $f_{X,Y}(x,y) = f_{Y \lvert X=x}(y \lvert x) \times f_X(x)$. In our problem at hand, the joint density function is:

\displaystyle \begin{aligned} f_{X,Y}(x,y)&=f_{Y \lvert X=x}(y \lvert x) \times f_X(x) \\&=\frac{1}{x} \times \alpha^2 \ x \ e^{-\alpha x} \\&=\alpha^2 \ e^{-\alpha x} \end{aligned}

As indicated above, the support of $f_{X,Y}(x,y)$ is the region $x>0$ and $0 (the region shaded green in the above figures).

Problem A-2

The unconditional density function of $X$ is $f_X(x)=\alpha^2 \ x \ e^{-\alpha x}$ (given above in the problem) is the density function of the sum of two independent exponential variables with the common density $f(x)=\alpha e^{-\alpha x}$ (see this blog post for the derivation using convolution method). Since $X$ is the independent sum of two identical exponential distributions, the mean and variance of $X$ is twice that of the same item of the exponential distribution. We have:

$\displaystyle E(X)=\frac{2}{\alpha}$

$\displaystyle Var(X)=\frac{2}{\alpha^2}$

Problem A-3

To find the marginal density of $Y$, for each applicable $y$, we need to sum out the $x$. According to the following figure, for each $y$, we sum out all $x$ values in a horizontal line such that $y (see the blue horizontal line).

Figure 3

Thus we have:

\displaystyle \begin{aligned} f_Y(y)&=\int_y^\infty f_{X,Y}(x,y) \ dy \ dx \\&=\int_y^\infty \alpha^2 \ e^{-\alpha x} \ dy \ dx \\&=\alpha \int_y^\infty \alpha \ e^{-\alpha x} \ dy \ dx \\&= \alpha e^{-\alpha y} \end{aligned}

Thus the marginal distribution of $Y$ is an exponential distribution. The mean and variance of $Y$ are:

$\displaystyle E(Y)=\frac{1}{\alpha}$

$\displaystyle Var(Y)=\frac{1}{\alpha^2}$

Problem A-4

The covariance of $X$ and $Y$ is defined as $Cov(X,Y)=E[(X-\mu_X) (Y-\mu_Y)]$, which is equivalent to:

$\displaystyle Cov(X,Y)=E(X Y)-\mu_X \mu_Y$

where $\mu_X=E(X)$ and $\mu_Y=E(Y)$. Knowing the joint density $f_{X,Y}(x,y)$, we can calculate $Cov(X,Y)$ directly. We have:

\displaystyle \begin{aligned} E(X Y)&=\int_0^\infty \int_0^x xy \ f_{X,Y}(x,y) \ dy \ dx \\&=\int_0^\infty \int_0^x xy \ \alpha^2 \ e^{-\alpha x} \ dy \ dx \\&=\int_0^\infty \frac{\alpha^2}{2} \ x^3 \ e^{-\alpha x} \ dy \ dx \\&= \frac{3}{\alpha^2} \int_0^\infty \frac{\alpha^4}{3!} \ x^{4-1} \ e^{-\alpha x} \ dy \ dx \\&= \frac{3}{\alpha^2} \end{aligned}

Note that the last integrand in the last integral in the above derivation is that of a Gamma distribution (hence the integral is 1.0). Now the covariance of $X$ and $Y$ is:

$\displaystyle Cov(X,Y)=\frac{3}{\alpha^2}-\frac{2}{\alpha} \frac{1}{\alpha}=\frac{1}{\alpha^2}$

The following is the calculation of the correlation coefficient:

\displaystyle \begin{aligned} \rho&=\frac{Cov(X,Y)}{\sigma_X \ \sigma_Y} = \frac{\displaystyle \frac{1}{\alpha^2}}{\displaystyle \frac{\sqrt{2}}{\alpha} \ \frac{1}{\alpha}} \\&=\frac{1}{\sqrt{2}} = 0.7071 \end{aligned}

Even without the calculation of $\rho$, we know that $X$ and $Y$ are positively and quite strongly correlated. The conditional distribution of $Y \lvert X=x$ is $U(0,x)$ which increases with $x$. The calculation of $Cov(X,Y)$ and $\rho$ confirms our observation.

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Problem B-1

$\displaystyle f_{X,Y}(x,y)=4 \ x^2$ where $x>0$, and $0.

Problem B-2

$\displaystyle E(X)=\frac{4}{5}$
$\displaystyle Var(X)=\frac{2}{75}$

Problem B-3

$\displaystyle f_Y(y)=\frac{4}{3} \ (1- y^3)$

$\displaystyle E(Y)=\frac{2}{5}$

$\displaystyle Var(Y)=\frac{14}{225}$

Problem B-4

$\displaystyle Cov(X,Y)=\frac{1}{75}$

$\displaystyle \rho = \frac{\sqrt{3}}{2 \sqrt{7}}=0.327327$

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## An Example on Calculating Covariance

Probem 1
Let $X$ be the value of one roll of a fair die. If the value of the die is $x$, we are given that $Y \lvert X=x$ has a binomial distribution with $n=x$ and $p=\frac{1}{4}$ (we use the notation $Y \lvert X=x \sim \text{binom}(x,\frac{1}{4})$).

1. Compute the mean and variance of $X$.
2. Compute the mean and variance of $Y$.
3. Compute the covariance $Cov(X,Y)$ and the correlation coefficient $\rho$.

Probem 2
Let $X$ be the value of one roll of a fair die. If the value of the die is $x$, we are given that $Y \lvert X=x$ has a binomial distribution with $n=x$ and $p=\frac{1}{2}$ (we use the notation $Y \lvert X=x \sim \text{binom}(x,\frac{1}{2})$).

1. Compute the mean and variance of $X$.
2. Compute the mean and variance of $Y$.
3. Compute the covariance $Cov(X,Y)$ and the correlation coefficient $\rho$.

Problem 2 is left as exercise.

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Discussion of Problem 1

The joint variables $X$ and $Y$ are identical to the ones in this previous post. However, we do not plan on following the approach in the previous, which is to first find the probability functions for the joint distribution and then the marginal distribution of $Y$. The calculation of covariance in Problem 1.3 can be very tedious by taking this approach.

Problem 1.1
We start with the easiest part, which is the random variable $X$ (the roll of the die). The variance is computed by $Var(X)=E(X^2)-E(X)^2$.

$\displaystyle (1) \ \ \ \ \ E(X)=\frac{1}{6} \biggl[1+2+3+4+5+6 \biggr]=\frac{21}{6}=3.5$

$\displaystyle (2) \ \ \ \ \ E(X^2)=\frac{1}{6} \biggl[1^2+2^2+3^2+4^2+5^2+6^2 \biggr]=\frac{91}{6}$

$\displaystyle (3) \ \ \ \ \ Var(X)=\frac{91}{6}-\biggl[\frac{21}{6}\biggr]^2=\frac{105}{36}=\frac{35}{12}$

Problem 1.2

We now compute the mean and variance of $Y$. The calculation of finding the joint distribution and then finding the marginal distribution of $Y$ is tedious and has been done in this previous post. We do not take this approach here. Instead, we find the unconditional mean $E(Y)$ by weighting the conditional mean $E(Y \lvert X=x)$. The weights are the probabilities $P(X=x)$. The following is the idea.

\displaystyle \begin{aligned}(4) \ \ \ \ \ E(Y)&=E_X[E(Y \lvert X=x)] \\&= E(Y \lvert X=1) \times P(X=1) \\&+ E(Y \lvert X=2) \times P(X=2)\\&+ E(Y \lvert X=3) \times P(X=3) \\&+ E(Y \lvert X=4) \times P(X=4) \\&+E(Y \lvert X=5) \times P(X=5) \\&+E(Y \lvert X=6) \times P(X=6) \end{aligned}

We have $P(X=x)=\frac{1}{6}$ for each $x$. Before we do the weighting, we need to have some items about the conditional distribution $Y \lvert X=x$. Since $Y \lvert X=x$ has a binomial distribution, we have:

$\displaystyle (5) \ \ \ \ \ E(Y \lvert X=x)=\frac{1}{4} \ x$

$\displaystyle (6) \ \ \ \ \ Var(Y \lvert X=x)=\frac{1}{4} \ \frac{3}{4} \ x=\frac{3}{16} \ x$

For any random variable $W$, $Var(W)=E(W^2)-E(W)^2$ and $E(W^2)=Var(W)+E(W)^2$. The following is the second moment of $Y \lvert X=x$, which is needed in calculating the unconditional variance $Var(Y)$.

\displaystyle \begin{aligned}(7) \ \ \ \ \ E(Y^2 \lvert X=x)&=\frac{3}{16} \ x+\biggl[\frac{1}{4} \ x \biggr]^2 \\&=\frac{3x}{16}+\frac{x^2}{16} \\&=\frac{3x+x^2}{16} \end{aligned}

We can now do the weighting to get the items of the variable $Y$.

\displaystyle \begin{aligned}(8) \ \ \ \ \ E(Y)&=\frac{1}{6} \biggl[\frac{1}{4} +\frac{2}{4}+\frac{3}{4}+ \frac{4}{4}+\frac{5}{4}+\frac{6}{4}\biggr] \\&=\frac{7}{8} \\&=0.875 \end{aligned}

\displaystyle \begin{aligned}(9) \ \ \ \ \ E(Y^2)&=\frac{1}{6} \biggl[\frac{3(1)+1^2}{16} +\frac{3(2)+2^2}{16}+\frac{3(3)+3^2}{16} \\&+ \frac{3(4)+4^2}{16}+\frac{3(5)+5^2}{16}+\frac{3(6)+6^2}{16}\biggr] \\&=\frac{154}{96} \\&=\frac{77}{48} \end{aligned}

\displaystyle \begin{aligned}(10) \ \ \ \ \ Var(Y)&=E(Y^2)-E(Y)^2 \\&=\frac{77}{48}-\biggl[\frac{7}{8}\biggr]^2 \\&=\frac{161}{192} \\&=0.8385 \end{aligned}

Problem 1.3

The following is the definition of covariance of $X$ and $Y$:

$\displaystyle (11) \ \ \ \ \ Cov(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]$

where $\mu_X=E(X)$ and $\mu_Y=E(Y)$.

The definition $(11)$ can be simplified as:

$\displaystyle (12) \ \ \ \ \ Cov(X,Y)=E[XY]-E[X] E[Y]$

To compute $E[XY]$, we can use the joint probability function of $X$ and $Y$ to compute this expectation. But this is tedious. Anyone who wants to try can go to this previous post to obtain the joint distribution.

Note that the conditional mean $E(Y \lvert X=x)=\frac{x}{4}$ is a linear function of $x$. It is a well known result in probability and statistics that whenever a conditional mean $E(Y \lvert X=x)$ is a linear function of $x$, the conditional mean can be written as:

$\displaystyle (13) \ \ \ \ \ E(Y \lvert X=x)=\mu_Y+\rho \ \frac{\sigma_Y}{\sigma_X} \ (x-\mu_X)$

where $\mu$ is the mean of the respective variable, $\sigma$ is the standard deviation of the respective variable and $\rho$ is the correlation coefficient. The following relates the correlation coefficient with the covariance.

$\displaystyle (14) \ \ \ \ \ \rho=\frac{Cov(X,Y)}{\sigma_X \ \sigma_Y}$

Comparing $(5)$ and $(13)$, we have $\displaystyle \rho \frac{\sigma_Y}{\sigma_X}=\frac{1}{4}$ and

$\displaystyle (15) \ \ \ \ \ \rho = \frac{\sigma_X}{4 \ \sigma_Y}$

Equating $(14)$ and $(15)$, we have $Cov(X,Y)=\frac{\sigma_X^2}{4}$. Thus we deduce that $Cov(X,Y)$ is one-fourth of the variance of $X$. Using $(3)$, we have:

$\displaystyle (16) \ \ \ \ \ Cov(X,Y) = \frac{1}{4} \times \frac{35}{12}=\frac{35}{48}=0.72917$

Plug in all the items of $(3)$, $(10)$, and $(16)$ into $(14)$, we obtained $\rho=0.46625$. Both $\rho$ and $Cov(X,Y)$ are positive, an indication that both variables move together. When one increases, the other variable also increases. Thus makes sense based on the definition of the variables. For example, when the value of the die is large, the number of trials of $Y$ is greater (hence a larger mean).

## An Example of a Joint Distribution

Probem 1
Let $X$ be the value of one roll of a fair die. If the value of the die is $x$, we are given that $Y \lvert X=x$ has a binomial distribution with $n=x$ and $p=\frac{1}{4}$ (we use the notation $Y \lvert X=x \sim \text{binom}(x,\frac{1}{4})$).

1. Discuss how the joint probability function $P[X=x,Y=y]$ is computed for $x=1,2,3,4,5,6$ and $y=0,1, \cdots, x$.
2. Compute the conditional binomial distributions $Y \lvert X=x$ where $x=1,2,3,4,5,6$.
3. Compute the marginal probability function of $Y$ and the mean and variance of $Y$.
4. Compute $P(X=x \lvert Y=y)$ for all applicable $x$ and $y$.

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Discussion of Problem 1

Problem 2 is found at the end of the post.

Problem 1.1
This is an example of a joint distribution that is constructed from taking product of conditional distributions and a marginial distribution. The marginal distribution of $X$ is a uniform distribution on the set $\left\{1,2,3,4,5,6 \right\}$ (rolling a fiar die). Conditional of $X=x$, $Y$ has a binomial distribution $\text{binom}(x,\frac{1}{4})$. Think of the conditional variable of $Y \lvert X=x$ as tossing a coin $x$ times where the probability of a head is $p=\frac{1}{4}$. The following is the sample space of the joint distribution of $X$ and $Y$.

Figure 1

The joint probability function of $X$ and $Y$ may be written as:

$\displaystyle (1) \ \ \ \ \ P(X=x,Y=y)=P(Y=y \lvert X=x) \times P(X=x)$

Thus the probability at each point in Figure 1 is the product of $P(X=x)$, which is $\frac{1}{6}$, with the conditional probability $P(Y=y \lvert X=x)$, which is binomial. For example, the following diagram and equation demonstrate the calculation of $P(X=4,Y=3)$

Figure 2

\displaystyle \begin{aligned}(1a) \ \ \ \ \ P(X=4,Y=3)&=P(Y=3 \lvert X=4) \times P(X=4) \\&=\binom{4}{3} \biggl[\frac{1}{4}\biggr]^3 \biggl[\frac{3}{4}\biggr]^1 \times \frac{1}{6} \\&=\frac{12}{256} \end{aligned}

Problem 1.2
The following shows the calculation of the binomial distributions.

\displaystyle \begin{aligned} (2) \ \ \ Y \lvert X=1 \ \ \ \ \ &P(Y=0 \lvert X=1)=\frac{3}{4} \\&P(Y=1 \lvert X=1)=\frac{1}{4} \end{aligned}

\displaystyle \begin{aligned} (3) \ \ \ Y \lvert X=2 \ \ \ \ \ &P(Y=0 \lvert X=2)=\binom{2}{0} \biggl(\frac{1}{4}\biggr)^0 \biggl(\frac{3}{4}\biggr)^2=\frac{9}{16} \\&P(Y=1 \lvert X=2)=\binom{2}{1} \biggl(\frac{1}{4}\biggr)^1 \biggl(\frac{3}{4}\biggr)^1=\frac{6}{16} \\&P(Y=2 \lvert X=2)=\binom{2}{2} \biggl(\frac{1}{4}\biggr)^2 \biggl(\frac{3}{4}\biggr)^0=\frac{1}{16} \end{aligned}

\displaystyle \begin{aligned} (4) \ \ \ Y \lvert X=3 \ \ \ \ \ &P(Y=0 \lvert X=3)=\binom{3}{0} \biggl(\frac{1}{4}\biggr)^0 \biggl(\frac{3}{4}\biggr)^3=\frac{27}{64} \\&P(Y=1 \lvert X=3)=\binom{3}{1} \biggl(\frac{1}{4}\biggr)^1 \biggl(\frac{3}{4}\biggr)^2=\frac{27}{64} \\&P(Y=2 \lvert X=3)=\binom{3}{2} \biggl(\frac{1}{4}\biggr)^2 \biggl(\frac{3}{4}\biggr)^1=\frac{9}{64} \\&P(Y=3 \lvert X=3)=\binom{3}{3} \biggl(\frac{1}{4}\biggr)^3 \biggl(\frac{3}{4}\biggr)^0=\frac{1}{64} \end{aligned}

\displaystyle \begin{aligned} (5) \ \ \ Y \lvert X=4 \ \ \ \ \ &P(Y=0 \lvert X=4)=\binom{4}{0} \biggl(\frac{1}{4}\biggr)^0 \biggl(\frac{3}{4}\biggr)^4=\frac{81}{256} \\&P(Y=1 \lvert X=4)=\binom{4}{1} \biggl(\frac{1}{4}\biggr)^1 \biggl(\frac{3}{4}\biggr)^3=\frac{108}{256} \\&P(Y=2 \lvert X=4)=\binom{4}{2} \biggl(\frac{1}{4}\biggr)^2 \biggl(\frac{3}{4}\biggr)^2=\frac{54}{256} \\&P(Y=3 \lvert X=4)=\binom{4}{3} \biggl(\frac{1}{4}\biggr)^3 \biggl(\frac{3}{4}\biggr)^1=\frac{12}{256} \\&P(Y=4 \lvert X=4)=\binom{4}{4} \biggl(\frac{1}{4}\biggr)^4 \biggl(\frac{3}{4}\biggr)^0=\frac{1}{256} \end{aligned}

\displaystyle \begin{aligned} (6) \ \ \ Y \lvert X=5 \ \ \ \ \ &P(Y=0 \lvert X=5)=\binom{5}{0} \biggl(\frac{1}{4}\biggr)^0 \biggl(\frac{3}{4}\biggr)^5=\frac{243}{1024} \\&P(Y=1 \lvert X=5)=\binom{5}{1} \biggl(\frac{1}{4}\biggr)^1 \biggl(\frac{3}{4}\biggr)^4=\frac{405}{1024} \\&P(Y=2 \lvert X=5)=\binom{5}{2} \biggl(\frac{1}{4}\biggr)^2 \biggl(\frac{3}{4}\biggr)^3=\frac{270}{1024} \\&P(Y=3 \lvert X=5)=\binom{5}{3} \biggl(\frac{1}{4}\biggr)^3 \biggl(\frac{3}{4}\biggr)^2=\frac{90}{1024} \\&P(Y=4 \lvert X=5)=\binom{5}{4} \biggl(\frac{1}{4}\biggr)^4 \biggl(\frac{3}{4}\biggr)^1=\frac{15}{1024} \\&P(Y=5 \lvert X=5)=\binom{5}{5} \biggl(\frac{1}{4}\biggr)^5 \biggl(\frac{3}{4}\biggr)^0=\frac{1}{1024} \end{aligned}

\displaystyle \begin{aligned} (7) \ \ \ Y \lvert X=6 \ \ \ \ \ &P(Y=0 \lvert X=6)=\binom{6}{0} \biggl(\frac{1}{4}\biggr)^0 \biggl(\frac{3}{4}\biggr)^6=\frac{729}{4096} \\&P(Y=1 \lvert X=6)=\binom{6}{1} \biggl(\frac{1}{4}\biggr)^1 \biggl(\frac{3}{4}\biggr)^5=\frac{1458}{4096} \\&P(Y=2 \lvert X=6)=\binom{6}{2} \biggl(\frac{1}{4}\biggr)^2 \biggl(\frac{3}{4}\biggr)^4=\frac{1215}{4096} \\&P(Y=3 \lvert X=6)=\binom{6}{3} \biggl(\frac{1}{4}\biggr)^3 \biggl(\frac{3}{4}\biggr)^3=\frac{540}{4096} \\&P(Y=4 \lvert X=6)=\binom{6}{4} \biggl(\frac{1}{4}\biggr)^4 \biggl(\frac{3}{4}\biggr)^2=\frac{135}{4096} \\&P(Y=5 \lvert X=6)=\binom{6}{5} \biggl(\frac{1}{4}\biggr)^5 \biggl(\frac{3}{4}\biggr)^1=\frac{18}{4096} \\&P(Y=6 \lvert X=6)=\binom{6}{6} \biggl(\frac{1}{4}\biggr)^6 \biggl(\frac{3}{4}\biggr)^0=\frac{1}{4096} \end{aligned}

Problem 1.3
To find the marginal probability $P(Y=y)$, we need to sum $P(X=x,Y=y)$ over all $x$. For example, $P(Y=2)$ is the sum of $P(X=x,Y=2)$ for all $x=2,3,4,5,6$. See the following diagram

Figure 3

As indicated in $(1)$, each $P(X=x,Y=2)$ is the product of a conditional probability $P(Y=y \lvert X=x)$ and $P(X=x)=\frac{1}{6}$. Thus the probability indicated in Figure 3 can be translated as:

\displaystyle \begin{aligned}(8) \ \ \ \ \ P(Y=2)&=\sum \limits_{x=2}^6 P(Y=2 \lvert X=x) P(X=x) \end{aligned}

We now begin the calculation.

\displaystyle \begin{aligned}(9) \ \ \ \ \ P(Y=0)&=\sum \limits_{x=1}^6 P(Y=0 \lvert X=x) P(X=x) \\&=\frac{1}{6} \biggl[ \frac{3}{4}+\frac{9}{16}+\frac{27}{64} \\&+ \ \ \ \frac{81}{256}+\frac{243}{1024}+\frac{729}{4096} \biggr] \\&=\frac{10101}{24576} \end{aligned}

\displaystyle \begin{aligned}(10) \ \ \ \ \ P(Y=1)&=\sum \limits_{x=1}^6 P(Y=1 \lvert X=x) P(X=x) \\&=\frac{1}{6} \biggl[ \frac{1}{4}+\frac{6}{16}+\frac{27}{64} \\&+ \ \ \ \frac{108}{256}+\frac{405}{1024}+\frac{1458}{4096} \biggr] \\&=\frac{9094}{24576} \end{aligned}

\displaystyle \begin{aligned}(11) \ \ \ \ \ P(Y=2)&=\sum \limits_{x=2}^6 P(Y=2 \lvert X=x) P(X=x) \\&=\frac{1}{6} \biggl[ \frac{1}{16}+\frac{9}{64} \\&+ \ \ \ \frac{54}{256}+\frac{270}{1024}+\frac{1215}{4096} \biggr] \\&=\frac{3991}{24576} \end{aligned}

\displaystyle \begin{aligned}(12) \ \ \ \ \ P(Y=3)&=\sum \limits_{x=3}^6 P(Y=3 \lvert X=x) P(X=x) \\&=\frac{1}{6} \biggl[ \frac{1}{64} \\&+ \ \ \ \frac{12}{256}+\frac{90}{1024}+\frac{540}{4096} \biggr] \\&=\frac{1156}{24576} \end{aligned}

\displaystyle \begin{aligned}(13) \ \ \ \ \ P(Y=4)&=\sum \limits_{x=4}^6 P(Y=4 \lvert X=x) P(X=x) \\&=\frac{1}{6} \biggl[ \frac{1}{256}+\frac{15}{1024}+\frac{135}{4096} \biggr] \\&=\frac{211}{24576} \end{aligned}

\displaystyle \begin{aligned}(14) \ \ \ \ \ P(Y=5)&=\sum \limits_{x=5}^6 P(Y=5 \lvert X=x) P(X=x) \\&=\frac{1}{6} \biggl[ \frac{1}{1024}+\frac{18}{4096} \biggr] \\&=\frac{22}{24576} \end{aligned}

\displaystyle \begin{aligned}(15) \ \ \ \ \ P(Y=6)&=\sum \limits_{x=6}^6 P(Y=6 \lvert X=x) P(X=x) \\&=\frac{1}{6} \biggl[ \frac{1}{4096} \biggr] \\&=\frac{1}{24576} \end{aligned}

The following is the calculation of the mean and variance of $Y$.

\displaystyle \begin{aligned}(16) \ \ \ \ \ E(Y)&=\frac{10101}{24576} \times 0+\frac{9094}{24576} \times 1+\frac{3991}{24576} \times 2 \\&+ \ \ \ \ \frac{1156}{24576} \times 3+\frac{211}{24576} \times 4+\frac{22}{24576} \times 5 \\&+ \ \ \ \ \frac{1}{24576} \times 6 \\&=\frac{21504}{24576}\\&=0.875 \end{aligned}

\displaystyle \begin{aligned}(17) \ \ \ \ \ E(Y^2)&=\frac{10101}{24576} \times 0+\frac{9094}{24576} \times 1+\frac{3991}{24576} \times 2^2 \\&+ \ \ \ \ \frac{1156}{24576} \times 3^2+\frac{211}{24576} \times 4^2+\frac{22}{24576} \times 5^2 \\&+ \ \ \ \ \frac{1}{24576} \times 6^2 \\&=\frac{39424}{24576}\\&=\frac{77}{48} \end{aligned}

$\displaystyle (18) \ \ \ \ \ Var(Y)=\frac{77}{48}-0.875^2=\frac{161}{192}=0.8385$

Problem 1.4
The conditional probability $P(Y=y \lvert X=x)$ is easy to compute since it is a given that $Y$ is a binomial variable conditional on a value of $X$. Now we want to find the backward probability $P(X= x \lvert Y=y)$. Given the binomial observation is $Y=y$, what is the probability that the roll of the die is $X=x$? This is an application of the Bayes’ theorem. We can start by looking at Figure 3 once more.

Consider $P(X=x \lvert Y=2)$. In calculating this conditional probability, we only consider the 5 sample points encircled in Figure 3 and disregard all the other points. These 5 points become a new sample space if you will (this is the essence of conditional probability and conditional distribution). The sum of the joint probability $P(X=x,Y=y)$ for these 5 points is $P(Y=2)$, calculated in the previous step. The conditional probability $P(X=x \lvert Y=2)$ is simply the probability of one of these 5 points as a fraction of the total probability $P(Y=2)$. Thus we have:

\displaystyle \begin{aligned}(19) \ \ \ \ \ P(X=x \lvert Y=2)&=\frac{P(X=x,Y=2)}{P(Y=2)} \end{aligned}

We do not have to evaluate the components that go into $(19)$. As a practical matter, to find $P(X=x \lvert Y=2)$ is to take each of 5 probabilities shown in $(11)$ and evaluate it as a fraction of the total probability $P(Y=2)$. Thus we have:

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 2 \bold )$
\displaystyle \begin{aligned}(20a) \ \ \ \ \ P(X=2 \lvert Y=2)&=\frac{\displaystyle \frac{1}{6} \times \frac{1}{16}}{\displaystyle \frac{3991}{24576}} =\frac{256}{3991} \end{aligned}

\displaystyle \begin{aligned}(20b) \ \ \ \ \ P(X=3 \lvert Y=2)&=\frac{\displaystyle \frac{1}{6} \times \frac{9}{64}}{\displaystyle \frac{3991}{24576}} =\frac{576}{3991} \end{aligned}

\displaystyle \begin{aligned}(20c) \ \ \ \ \ P(X=4 \lvert Y=2)&=\frac{\displaystyle \frac{1}{6} \times \frac{54}{256}}{\displaystyle \frac{3991}{24576}} =\frac{864}{3991} \end{aligned}

\displaystyle \begin{aligned}(20d) \ \ \ \ \ P(X=5 \lvert Y=2)&=\frac{\displaystyle \frac{1}{6} \times \frac{270}{1024}}{\displaystyle \frac{3991}{24576}} =\frac{1080}{3991} \end{aligned}

\displaystyle \begin{aligned}(20e) \ \ \ \ \ P(X=6 \lvert Y=2)&=\frac{\displaystyle \frac{1}{6} \times \frac{1215}{4096}}{\displaystyle \frac{3991}{24576}} =\frac{1215}{3991} \end{aligned}

Here’s the rest of the Bayes’ calculation:

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 0 \bold )$
\displaystyle \begin{aligned}(21a) \ \ \ \ \ P(X=1 \lvert Y=0)&=\frac{\displaystyle \frac{1}{6} \times \frac{3}{4}}{\displaystyle \frac{10101}{24576}} =\frac{3072}{10101} \end{aligned}

\displaystyle \begin{aligned}(21b) \ \ \ \ \ P(X=2 \lvert Y=0)&=\frac{\displaystyle \frac{1}{6} \times \frac{9}{16}}{\displaystyle \frac{10101}{24576}} =\frac{2304}{10101} \end{aligned}

\displaystyle \begin{aligned}(21c) \ \ \ \ \ P(X=3 \lvert Y=0)&=\frac{\displaystyle \frac{1}{6} \times \frac{27}{64}}{\displaystyle \frac{10101}{24576}} =\frac{1728}{10101} \end{aligned}

\displaystyle \begin{aligned}(21d) \ \ \ \ \ P(X=4 \lvert Y=0)&=\frac{\displaystyle \frac{1}{6} \times \frac{81}{256}}{\displaystyle \frac{10101}{24576}} =\frac{1296}{10101} \end{aligned}

\displaystyle \begin{aligned}(21e) \ \ \ \ \ P(X=5 \lvert Y=0)&=\frac{\displaystyle \frac{1}{6} \times \frac{243}{1024}}{\displaystyle \frac{10101}{24576}} =\frac{972}{10101} \end{aligned}

\displaystyle \begin{aligned}(21f) \ \ \ \ \ P(X=6 \lvert Y=0)&=\frac{\displaystyle \frac{1}{6} \times \frac{729}{4096}}{\displaystyle \frac{10101}{24576}} =\frac{3729}{10101} \end{aligned}

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 1 \bold )$
\displaystyle \begin{aligned}(22a) \ \ \ \ \ P(X=1 \lvert Y=1)&=\frac{\displaystyle \frac{1}{6} \times \frac{1}{4}}{\displaystyle \frac{9094}{24576}} =\frac{1024}{9094} \end{aligned}

\displaystyle \begin{aligned}(22b) \ \ \ \ \ P(X=2 \lvert Y=1)&=\frac{\displaystyle \frac{1}{6} \times \frac{6}{16}}{\displaystyle \frac{9094}{24576}} =\frac{1536}{9094} \end{aligned}

\displaystyle \begin{aligned}(22c) \ \ \ \ \ P(X=3 \lvert Y=1)&=\frac{\displaystyle \frac{1}{6} \times \frac{27}{64}}{\displaystyle \frac{9094}{24576}} =\frac{1728}{9094} \end{aligned}

\displaystyle \begin{aligned}(22d) \ \ \ \ \ P(X=4 \lvert Y=1)&=\frac{\displaystyle \frac{1}{6} \times \frac{108}{256}}{\displaystyle \frac{9094}{24576}} =\frac{1728}{9094} \end{aligned}

\displaystyle \begin{aligned}(22e) \ \ \ \ \ P(X=5 \lvert Y=1)&=\frac{\displaystyle \frac{1}{6} \times \frac{405}{1024}}{\displaystyle \frac{9094}{24576}} =\frac{1620}{9094} \end{aligned}

\displaystyle \begin{aligned}(22f) \ \ \ \ \ P(X=6 \lvert Y=1)&=\frac{\displaystyle \frac{1}{6} \times \frac{1458}{4096}}{\displaystyle \frac{9094}{24576}} =\frac{1458}{9094} \end{aligned}

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 2 \bold )$ done earlier

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 3 \bold )$
\displaystyle \begin{aligned}(23a) \ \ \ \ \ P(X=3 \lvert Y=3)&=\frac{\displaystyle \frac{1}{6} \times \frac{1}{64}}{\displaystyle \frac{1156}{24576}} =\frac{64}{1156} \end{aligned}

\displaystyle \begin{aligned}(23b) \ \ \ \ \ P(X=4 \lvert Y=3)&=\frac{\displaystyle \frac{1}{6} \times \frac{12}{256}}{\displaystyle \frac{1156}{24576}} =\frac{192}{1156} \end{aligned}

\displaystyle \begin{aligned}(23c) \ \ \ \ \ P(X=5 \lvert Y=3)&=\frac{\displaystyle \frac{1}{6} \times \frac{90}{1024}}{\displaystyle \frac{1156}{24576}} =\frac{360}{1156} \end{aligned}

\displaystyle \begin{aligned}(23d) \ \ \ \ \ P(X=6 \lvert Y=3)&=\frac{\displaystyle \frac{1}{6} \times \frac{540}{4096}}{\displaystyle \frac{1156}{24576}} =\frac{540}{1156} \end{aligned}

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 4 \bold )$
\displaystyle \begin{aligned}(24a) \ \ \ \ \ P(X=4 \lvert Y=4)&=\frac{\displaystyle \frac{1}{6} \times \frac{1}{256}}{\displaystyle \frac{211}{24576}} =\frac{16}{211} \end{aligned}

\displaystyle \begin{aligned}(24b) \ \ \ \ \ P(X=5 \lvert Y=4)&=\frac{\displaystyle \frac{1}{6} \times \frac{15}{1024}}{\displaystyle \frac{211}{24576}} =\frac{60}{211} \end{aligned}

\displaystyle \begin{aligned}(24c) \ \ \ \ \ P(X=6 \lvert Y=4)&=\frac{\displaystyle \frac{1}{6} \times \frac{135}{4096}}{\displaystyle \frac{211}{24576}} =\frac{135}{211} \end{aligned}

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 5 \bold )$
\displaystyle \begin{aligned}(25a) \ \ \ \ \ P(X=5 \lvert Y=5)&=\frac{\displaystyle \frac{1}{6} \times \frac{1}{1024}}{\displaystyle \frac{22}{24576}} =\frac{4}{22} \end{aligned}

\displaystyle \begin{aligned}(25b) \ \ \ \ \ P(X=6 \lvert Y=5)&=\frac{\displaystyle \frac{1}{6} \times \frac{18}{1024}}{\displaystyle \frac{22}{24576}} =\frac{18}{22} \end{aligned}

Calculation of $\bold P \bold ( \bold X \bold = \bold x \bold \lvert \bold Y \bold = \bold 6 \bold )$
\displaystyle \begin{aligned}(26) \ \ \ \ \ P(X=6 \lvert Y=6)&=\frac{\displaystyle \frac{1}{6} \times \frac{1}{4096}}{\displaystyle \frac{1}{24576}} =1 \end{aligned}

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Probem 2
Let $X$ be the value of one roll of a fair die. If the value of the die is $x$, we are given that $Y \lvert X=x$ has a binomial distribution with $n=x$ and $p=\frac{1}{2}$ (we use the notation $Y \lvert X=x \sim \text{binom}(x,\frac{1}{2})$).

1. Discuss how the joint probability function $P[X=x,Y=y]$ is computed for $x=1,2,3,4,5,6$ and $y=0,1, \cdots, x$.
2. Compute the conditional binomial distributions $Y \lvert X=x$ where $x=1,2,3,4,5,6$.
3. Compute the marginal probability function of $Y$ and the mean and variance of $Y$.
4. Compute $P(X=x \lvert Y=y)$ for all applicable $x$ and $y$.

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Problem 2.3

\displaystyle \begin{aligned} P(Y=y): \ \ \ \ &P(Y=0)=\frac{63}{384} \\&\text{ } \\&P(Y=1)=\frac{120}{384} \\&\text{ } \\&P(Y=2)=\frac{99}{384} \\&\text{ } \\&P(Y=3)=\frac{64}{384} \\&\text{ } \\&P(Y=4)=\frac{29}{384} \\&\text{ } \\&P(Y=5)=\frac{8}{384} \\&\text{ } \\&P(Y=6)=\frac{1}{384} \end{aligned}

$\displaystyle E(Y)=\frac{7}{4}=1.75$

$\displaystyle Var(Y)=\frac{77}{48}$

Problem 2.4

\displaystyle \begin{aligned} P(X=x \lvert Y=0): \ \ \ \ &P(X=1 \lvert Y=0)=\frac{32}{63} \\&\text{ } \\&P(X=2 \lvert Y=0)=\frac{16}{63} \\&\text{ } \\&P(X=3 \lvert Y=0)=\frac{8}{63} \\&\text{ } \\&P(X=4 \lvert Y=0)=\frac{4}{63} \\&\text{ } \\&P(X=5 \lvert Y=0)=\frac{2}{63} \\&\text{ } \\&P(X=6 \lvert Y=0)=\frac{1}{63} \end{aligned}

\displaystyle \begin{aligned} P(X=x \lvert Y=1): \ \ \ \ &P(X=1 \lvert Y=1)=\frac{32}{120} \\&\text{ } \\&P(X=2 \lvert Y=1)=\frac{32}{120} \\&\text{ } \\&P(X=3 \lvert Y=1)=\frac{24}{120} \\&\text{ } \\&P(X=4 \lvert Y=1)=\frac{16}{120} \\&\text{ } \\&P(X=5 \lvert Y=1)=\frac{10}{120} \\&\text{ } \\&P(X=6 \lvert Y=1)=\frac{6}{120} \end{aligned}

\displaystyle \begin{aligned} P(X=x \lvert Y=2): \ \ \ \ &P(X=2 \lvert Y=2)=\frac{16}{99} \\&\text{ } \\&P(X=3 \lvert Y=2)=\frac{24}{99} \\&\text{ } \\&P(X=4 \lvert Y=2)=\frac{24}{99} \\&\text{ } \\&P(X=5 \lvert Y=2)=\frac{20}{99} \\&\text{ } \\&P(X=6 \lvert Y=2)=\frac{15}{99} \end{aligned}

\displaystyle \begin{aligned} P(X=x \lvert Y=3): \ \ \ \ &P(X=3 \lvert Y=3)=\frac{8}{64} \\&\text{ } \\&P(X=4 \lvert Y=3)=\frac{16}{64} \\&\text{ } \\&P(X=5 \lvert Y=3)=\frac{20}{64} \\&\text{ } \\&P(X=6 \lvert Y=3)=\frac{20}{64} \end{aligned}

\displaystyle \begin{aligned} P(X=x \lvert Y=4): \ \ \ \ &P(X=4 \lvert Y=4)=\frac{4}{29} \\&\text{ } \\&P(X=5 \lvert Y=4)=\frac{10}{29} \\&\text{ } \\&P(X=6 \lvert Y=4)=\frac{15}{29} \end{aligned}

\displaystyle \begin{aligned} P(X=x \lvert Y=5): \ \ \ \ &P(X=5 \lvert Y=5)=\frac{2}{8} \\&\text{ } \\&P(X=6 \lvert Y=5)=\frac{6}{8} \end{aligned}

\displaystyle \begin{aligned} P(X=x \lvert Y=6): \ \ \ \ &P(X=6 \lvert Y=6)=1 \end{aligned}