Chapter 10

Consider the two discrete random variables \(X\) and \(Y\) with joint distribution derived in Exercise 9.2: $$ \begin{array}{ccccc} \hline \hline & {a} & \\ { 2 - 4 } b & 0 & 1 & 2 & \mathrm{P}(Y=b) \\ \hline-1 & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 2 \\ 1 & 0 & 1 / 2 & 0 & 1 / 2 \\ \hline \mathrm{P}(X=a) & 1 / 6 & 2 / 3 & 1 / 6 & 1 \\ \hline \hline \end{array} $$ a. Determine \(\mathrm{E}[X Y]\). b. Note that \(X\) and \(Y\) are dependent. Show that \(X\) and \(Y\) are uncorrelated. c. Determine \(\operatorname{Var}(X+Y)\). d. Determine \(\operatorname{Var}(X-Y)\).

Consider the joint probability distribution of the discrete random variables \(X\) and \(Y\) from the Melencolia Exercise 9.1. Compute \(\operatorname{Cov}(X, Y)\). $$ \begin{array}{ccccc} \hline \hline & {a} \\ b & 1 & 2 & 3 & 4 \\ { 2 - 5 } & 16 / 136 & 3 / 136 & 2 / 136 & 13 / 136 \\ 2 & 5 / 136 & 10 / 136 & 11 / 136 & 8 / 136 \\ 3 & 9 / 136 & 6 / 136 & 7 / 136 & 12 / 136 \\ 4 & 4 / 136 & 15 / 136 & 14 / 136 & 1 / 136 \\ \hline \hline \end{array} $$

Suppose \(X\) and \(Y\) are discrete random variables taking values \(c-1, c\), and \(c+1\). The following is given about the joint and marginal distributions: $$ \begin{array}{ccccc} \hline & {3}{c}{a} & \\ b & c-1 & c & c+1 & \mathrm{P}(Y=b) \\ \hline c-1 & 2 / 45 & 9 / 45 & 4 / 45 & 1 / 3 \\ c & 7 / 45 & 5 / 45 & 3 / 45 & 1 / 3 \\ c+1 & 6 / 45 & 1 / 45 & 8 / 45 & 1 / 3 \\ \hline \mathrm{P}(X=a) & 1 / 3 & 1 / 3 & 1 / 3 & 1 \\ \hline \end{array} $$ a. Take \(c=0\) and compute the expectation of \(X\) and of \(Y\) and the covariance between \(X\) and \(Y\). b. Show that \(X\) and \(Y\) are uncorrelated, no matter what the value of \(c\) is. Hint: one could compute \(\operatorname{Cov}(X, Y)\), but there is a short solution using the rule on the covariance under change of units (see page 141 ) together with part a. c. Are \(X\) and \(Y\) independent?

Consider the joint distribution of Quick exercise \(9.2\) and take \(\varepsilon\) fixed between \(-1 / 4\) and \(1 / 4\) : $$ \begin{array}{cccc} & {b} & \\ a & 0 & 1 & p_{X}(a) \\ \hline 0 & 1 / 4-\varepsilon & 1 / 4+\varepsilon & 1 / 2 \\ 1 & 1 / 4+\varepsilon & 1 / 4-\varepsilon & 1 / 2 \\ \hline D_{Y}(b) & 1 / 2 & 1 / 2 & 1 \end{array} $$ a. Take \(\varepsilon=1 / 8\) and compute \(\operatorname{Cov}(X, Y)\). b. Take \(\varepsilon=1 / 8\) and compute \(\rho(X, Y)\). c. For which values of \(\varepsilon\) is \(\rho(X, Y)\) equal to \(-1,0\), or 1 ?

Suppose the blood of 1000 persons has to be tested to see which ones are infected by a (rare) disease. Suppose that the probability that the test is positive is \(p=0.001\). The obvious way to proceed is to test each person, which results in a total of 1000 tests. An alternative procedure is the following. Distribute the blood of the 1000 persons over 25 groups of size 40 , and mix half of the blood of each of the 40 persons with that of the others in each group. Now test the aggregated blood sample of each group: when the test is negative no one in that group has the disease; when the test is positive, at least one person in the group has the disease, and one will test the other half of the blood of all 40 persons of that group separately. In total, that gives 41 tests for that group. Let \(X_{i}\) be the total number of tests one has to perform for the \(i\) th group using this alternative procedure. a. Describe the probability distribution of \(X_{i}\), i.e., list the possible values it takes on and the corresponding probabilities. b. What is the expected number of tests for the \(i\) th group? What is the expected total number of tests? What do you think of this alternative procedure for blood testing?

Consider the variables \(X\) and \(Y\) from the example in Section \(9.2\) with joint probability density $$ f(x, y)=\frac{2}{75}\left(2 x^{2} y+x y^{2}\right) \quad \text { for } 0 \leq x \leq 3 \text { and } 1 \leq y \leq 2 $$ and marginal probability densities $$ \begin{array}{ll} f_{X}(x)=\frac{2}{225}\left(9 x^{2}+7 x\right) & \text { for } 0 \leq x \leq 3 \\\ f_{Y}(y)=\frac{1}{25}\left(3 y^{2}+12 y\right) & \text { for } 1 \leq y \leq 2 \end{array} $$ a. Compute \(\mathrm{E}[X], \mathrm{E}[Y]\), and \(\mathrm{E}[X+Y]\). b. Compute \(\mathrm{E}\left[X^{2}\right], \mathrm{E}\left[Y^{2}\right], \mathrm{E}[X Y]\), and \(\mathrm{E}\left[(X+Y)^{2}\right]\), c. Compute \(\operatorname{Var}(X+Y), \operatorname{Var}(X)\), and \(\operatorname{Var}(Y)\) and check that \(\operatorname{Var}(X+Y) \neq\) \(\operatorname{Var}(X)+\operatorname{Var}(Y)\)

Recall the relation between degrees Celsius and degrees Fahrenheit degrees Fahrenheit \(=\frac{9}{5} \cdot\) degrees Celsius \(+32\). Let \(X\) and \(Y\) be the average daily temperatures in degrees Celsius in Amsterdam and Antwerp. Suppose that \(\operatorname{Cov}(X, Y)=3\) and \(\rho(X, Y)=0.8\). Let \(T\) and \(S\) be the same temperatures in degrees Fahrenheit. Compute \(\operatorname{Cov}(T, S)\) and \(\rho(T, S)\).

Consider the independent random variables \(H\) and \(R\) from the vase example, with a \(U(25,35)\) and a \(U(7.5,12.5)\) distribution. Compute \(\mathrm{E}[H]\) and \(\mathrm{E}\left[R^{2}\right]\) and check that \(\mathrm{E}[V]=\pi \mathrm{E}[H] \mathrm{E}\left[R^{2}\right]\).

Let \(X\) and \(Y\) be two random variables and let \(r, s, t\), and \(u\) be arbitrary real numbers. a. Derive from the definition that \(\operatorname{Cov}(X+s, Y+u)=\operatorname{Cov}(X, Y)\). b. Derive from the definition that \(\operatorname{Cov}(r X, t Y)=r t \operatorname{Cov}(X, Y)\). c. Combine parts a and \(\mathbf{b}\) to show \(\operatorname{Cov}(r X+s, t Y+u)=r t \operatorname{Cov}(X, Y)\).

Let \(X\) and \(Y\) be random variables. a. Express \(\operatorname{Cov}(X, X+Y)\) in terms of \(\operatorname{Var}(X)\) and \(\operatorname{Cov}(X, Y)\). b. Are \(X\) and \(X+Y\) positively correlated, uncorrelated, or negatively correlated, or can anything happen? c. Same question as in part \(\mathbf{b}\), but now assume that \(X\) and \(Y\) are uncorrelated.

Consider a vase containing balls numbered \(1,2, \ldots, N\). We draw \(n\) balls without replacement from the vase. Each ball is selected with equal probability, i.e., in the first draw each ball has probability \(1 / N\), in the second draw each of the \(N-1\) remaining balls has probability \(1 /(N-1)\), and so on. For \(i=1,2, \ldots, n\), let \(X_{i}\) denote the number on the ball in the \(i\) th draw. From Exercise \(9.18\) we know that the variance of \(X_{i}\) equals $$ \operatorname{Var}\left(X_{i}\right)=\frac{1}{12}(N-1)(N+1) $$ Show that $$ \operatorname{Cov}\left(X_{1}, X_{2}\right)=-\frac{1}{12}(N+1) $$ Before you do the exercise: why do you think the covariance is negative? Hint: use \(\operatorname{Var}\left(X_{1}+X_{2}+\cdots+X_{N}\right)=0\) (why?), and apply Exercise \(10.17\).

Derive the alternative expression for the covariance: \(\operatorname{Cov}(X, Y)=\) \(\mathrm{E}[X Y]-\mathrm{E}[X] \mathrm{E}[Y] .\)

Determine \(\rho\left(U, U^{2}\right)\) when \(U\) has a \(U(0, a)\) distribution. Here \(a\) is a positive number.