# Calculating bivariate normal probabilities

This post extends the discussion of the bivariate normal distribution started in this post from a companion blog. Practice problems are given in the next post.

Suppose that the continuous random variables $X$ and $Y$ follow a bivariate normal distribution with parameters $\mu_X$, $\sigma_X$, $\mu_Y$, $\sigma_Y$ and $\rho$. What to make of these five parameters? According to the previous post, we know that

• $\mu_X$ and $\sigma_X$ are the mean and standard deviation of the marginal distribution of $X$,
• $\mu_Y$ and $\sigma_Y$ are the mean and standard deviation of the marginal distribution of $Y$,
• and finally $\rho$ is the correlation coefficient of $X$ and $Y$.

So the five parameters of a bivariate normal distribution are the means and standard deviations of the two marginal distributions and the fifth parameter is the correlation coefficient that serves to connect $X$ and $Y$. If $\rho=0$, then $X$ and $Y$ are simply two independent normal distributions.

When calculating probabilities involving a bivariate normal distribution, keep in mind that both marginal distributions are normal. Furthermore, the conditional distribution of one variable given a value of the other is also normal. Much more can be said about the conditional distributions.

The conditional distribution of $Y$ given $X=x$ is usually denoted by $Y \lvert X=x$ or $Y \lvert x$. In additional to being a normal distribution, it has a mean that is a linear function of $x$ and has a variance that is constant (it does not matter what $x$ is, the variance is always the same). The linear conditional mean and constant variance are given by the following:

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

$\displaystyle Var[Y \lvert X=x]=\sigma_Y^2 \ (1-\rho^2)$

Similarly, the conditional distribution of $X$ given $Y=y$ is usually denoted by $X \lvert Y=y$ or $X \lvert y$. In additional to being a normal distribution, it has a mean that is a linear function of $x$ and has a variance that is constant. The linear conditional mean and constant variance are given by the following:

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

$\displaystyle Var[X \lvert Y=y]=\sigma_X^2 \ (1-\rho^2)$

The information about the conditional distribution of $Y$ on $X=x$ is identical to the information about the conditional distribution of $X$ on $Y=y$, except for the switching of $X$ and $Y$. An example is helpful.

Example 1
Suppose that the continuous random variables $X$ and $Y$ follow a bivariate normal distribution with parameters $\mu_X=10$, $\sigma_X=10$, $\mu_Y=20$, $\sigma_Y=5$ and $\rho=0.6$. The first two parameters are the mean and standard deviation of the marginal distribution of $X$. The next two parameters are the mean and standard deviation of the marginal distribution of $Y$. The parameter $\rho$ is the correlation coefficient of $X$ and $Y$. Both marginal distributions are normal.

Let’s focus on the conditional distribution of $Y$ given $X=x$. It is normally distributed. Its mean and variance are:

\displaystyle \begin{aligned} E[Y \lvert X=x]&=\mu_Y+\rho \ \frac{\sigma_Y}{\sigma_X} \ (x-\mu_X) \\&=20+0.6 \ \frac{5}{10} \ (x-10) \\&=20+0.3 \ (x-10) \\&=17+0.3 \ x \end{aligned}

$\displaystyle \sigma_{Y \lvert x}^2=Var[Y \lvert X=x]=\sigma_Y^2 (1-\rho^2)=25 \ (1-0.6^2)=16$

$\displaystyle \sigma_{Y \lvert x}=4$

The line $y=17+0.3 \ x$ is also called the least squares regression line. It gives the mean of the conditional distribution of $Y$ given $x$. Because $X$ and $Y$ are positively correlated, the least squares line has positive slope. In this case, the larger the $x$, the larger is the mean of $Y$. The standard deviation of $Y$ given $x$ is constant across all possible $x$ values.

With mean and standard deviation known, we can now compute normal probabilities. Suppose the realized value of $X$ is 25. Then the mean of $Y \lvert 25$ is $E[Y \lvert 25]=24.5$. The standard deviation, as indicated above, is 4. In fact, for any other $x$, the standard deviation of $Y \lvert x$ is also 4. Now calculate the probability $P[20. We first calculate it using a normal table found here.

\displaystyle \begin{aligned} P[20

Using a TI84+ calculator, $P[20. In contrast, the probability $P[20 is (using the table found here):

\displaystyle \begin{aligned} P[20

Using a TI84+ calculator, $P[20. Note that $P[20 is for the marginal distribution of $Y$. It is not conditioned on any realized value of $X$.

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