# Practice Problem Set 1 – Order Statistics

This post presents exercises on order statistics, reinforcing the concepts discussed in two blog posts in a companion blog on mathematical statistics.

The first blog post from the companion blog is an introduction to order statistics. That post presents the probability distributions of the order statistics, both individually and jointly. The second post presents basic examples illustrating how to calculate the order statistics.

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 Practice Problem 1-A A random sample of size 4 is drawn from a population that has a uniform distribution on the interval $(0,5)$. The resulting order statistics are $X_{(1)}$, $X_{(2)}$, $X_{(3)}$ and $X_{(4)}$. Determine the cumulative distribution function (CDF) of the 3rd order statistic $X_{(3)}$. Evaluate the probability $P(X_{(3)}>2)$.

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 Practice Problem 1-B As in Problem 1-A, a random sample of size 4 is drawn from a population that has a uniform distribution on the interval $(0,5)$. The resulting order statistics are $X_{(1)}$, $X_{(2)}$, $X_{(3)}$ and $X_{(4)}$. Evaluate the conditional probability $P(X_{(4)}>4 \lvert X_{(3)} >2)$.

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 Practice Problem 1-C The random sample $X_1,X_2,\cdots,X_9$ of size 9 is drawn from a population that has a uniform distribution on the interval $(0,10)$. Evaluate the mean and variance of the 7th order statistic $X_{(7)}$.

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 Practice Problem 1-D Suppose that the random sample $X_1,X_2$ of size 2 is drawn from a population that has an exponential distribution with mean 10. Let $X_{(1)}$ be the sample minimum and let $X_{(2)}$ be the sample maximum. Evaluate the conditional probability $P(X_{(1)}<5 \lvert X_{(2)} <10)$.

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 Practice Problem 1-E Suppose that $X_1,X_2,X_3$ is a random sample drawn from an exponential distribution with mean 10. The sample median here is the 2nd order statistic $X_{(2)}$. Evaluate the probability that the sample median is between 5 and 10.

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 Practice Problem 1-F Suppose that $X_1,X_2,X_3$ is a random sample drawn from a uniform distribution on the interval $(0,2)$. Let $R=X_{(3)}-X_{(1)}$ be the sample range. Determine the CDF of the sample range $R$. Evaluate the mean and variance of the sample range $R$.

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 Practice Problem 1-G As in Problem 1-F, $X_1,X_2,X_3$ is a random sample drawn from a uniform distribution on the interval $(0,2)$. Let $R=X_{(3)}-X_{(1)}$ be the sample range. The following relationship relates the variance of the sample range $R$ with the variances and covariance of $X_{(1)}$ and $X_{(3)}$. $Var(R)=Var(X_{(1)})+Var(X_{(3)})-2 \ Cov(X_{(1)},X_{(3)})$ Evaluate $Cov(X_{(1)},X_{(3)})$ using the above relationship. Evaluate the correlation coefficient $\rho$ of $X_{(1)}$ and $X_{(3)}$.

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 Practice Problem 1-H Suppose that $X_1,X_2,X_3$ is a random sample drawn from a uniform distribution on the interval $(0,5)$. Let $X_{(1)}$,$X_{(2)}$ and $X_{(3)}$ be the resulting order statistics. Determine the conditional density function of $X_{(3)}$ given that $X_{(1)}=x, X_{(2)}=y$ for all $0. What is the distribution indicated by the conditional density function? Evaluate the condition mean $E(X_{(3)} \lvert X_{(1)}=x, X_{(2)}=y)$ and the conditional variance $Var(X_{(3)} \lvert X_{(1)}=x, X_{(2)}=y)$.

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 Practice Problem 1-I Suppose that $X_1,X_2,X_3$ is a random sample drawn from an exponential distribution with mean $\theta$. Let $X_{(1)}$,$X_{(2)}$ and $X_{(3)}$ be the resulting order statistics. Determine the conditional density function of $X_{(3)}$ given that $X_{(1)}=x, X_{(2)}=y$ for all $0. What is the distribution indicated by the conditional density function? Evaluate the condition mean $E(X_{(3)} \lvert X_{(1)}=x, X_{(2)}=y)$ and the conditional variance $Var(X_{(3)} \lvert X_{(1)}=x, X_{(2)}=y)$.

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 Practice Problem 1-J Suppose that $X_1,X_2,X_3$ is a random sample drawn from a continuous distribution with density function $f(x)=2x$ where $0. Let the resulting order statistics be $X_{(1)}$, $X_{(2)}$ and $X_{(3)}$ where $X_{(1)}$ is the sample minimum, $X_{(2)}$ is the sample median and $X_{(3)}$ is the sample maximum. Evaluate the mean and variance of the sample minimum $X_{(1)}$. Evaluate the mean and variance of the sample maximum $X_{(3)}$.

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 Practice Problem 1-K As in Problem 1-J, suppose that $X_1,X_2,X_3$ is a random sample drawn from a continuous distribution with density function $f(x)=2x$ where $0. Let the resulting order statistics be $X_{(1)}$, $X_{(2)}$ and $X_{(3)}$ where $X_{(1)}$ is the sample minimum, $X_{(2)}$ is the sample median and $X_{(3)}$ is the sample maximum. Evaluate the covariance between $X_{(1)}$ and $X_{(3)}$. Evaluate the correlation between $X_{(1)}$ and $X_{(3)}$.

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Problem ………..Answer
1-A
• $\displaystyle F_{X_{(3)}}(x)=\frac{20}{625} \ x^3-\frac{3}{625} \ x^4$
• $\displaystyle \frac{513}{625}=0.8208$
1-B
• $\displaystyle \frac{337}{513}=0.65692$
1-C
• $E(X_{(7)}=7$
• $\displaystyle Var(X_{(7)}=\frac{21}{11}=1.9$
1-D
• $\displaystyle P(X_{(2)}<10)=(1-e^{-1})^2$
• $\displaystyle P(X_{(1)}<5 \lvert X_{(2)}<10)=\frac{1-3 e^{-1}+2 e^{-1.5}}{(1-e^{-1})^2}=0.8575$
1-E
• $\displaystyle 3 \ (e^{-1}-e^{-2})-2 \ (e^{-1.5}-e^{-3})=0.3509$
1-F
• $\displaystyle F_R(r)=\frac{3}{4} \ r^2-\frac{1}{4} \ r^3$
• $\displaystyle E(R)=1$
• $\displaystyle Var(R)=\frac{1}{5}$
1-G
• $\displaystyle Cov(X_{(1)},X_{(3)})=\frac{1}{20}$
• $\displaystyle \rho=\frac{1}{3}$
1-H
• For $0, $\displaystyle f_{X_{(3)} \lvert X_{(1)}=x,X_{(2)}=y}(z \lvert x, y)=\frac{1}{5-y} \ \ \ \ \ y
• This is a uniform distribution on $(y,5)$
• $\displaystyle E(X_{(3)} \lvert X_{(1)}=x,X_{(2)}=y)=\frac{1}{2} \ (y+5)$
• $\displaystyle Var(X_{(3)} \lvert X_{(1)}=x,X_{(2)}=y)=\frac{(5-y)^2}{12}$
1-I
• For $0, $\displaystyle f_{X_{(3)} \lvert X_{(1)}=x,X_{(2)}=y}(z \lvert x, y)=\frac{\frac{1}{\theta} e^{-z/\theta}}{e^{-y/\theta}} \ \ \ \ \ y
• This is an exponential distribution conditional on $z>y$
• $\displaystyle E(X_{(3)} \lvert X_{(1)}=x,X_{(2)}=y)=y+\theta$
• $\displaystyle Var(X_{(3)} \lvert X_{(1)}=x,X_{(2)}=y)=\theta^2$
1-J
• $\displaystyle E(X_{(1)})=\frac{16}{35}$
• $\displaystyle Var(X_{(1)})=\frac{201}{4900}$
• $\displaystyle E(X_{(3)})=\frac{6}{7}$
• $\displaystyle Var(X_{(3)})=\frac{3}{196}$
1-K
• $\displaystyle Cov(X_{(1)},X_{(3)})=\frac{2}{245}$
• $\displaystyle \rho=\frac{8}{\sqrt{603}}=0.326$

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