# 22.5. Exercises¶

**1.** A box contains two coins, one of which lands heads with chance \(p_1\) while the other lands heads with chance \(p_2\). One of the coins is picked at random and tossed \(n\) times. Find the expectation and variance of the number of heads.

**2.** A positive random variable \(V\) has expectation \(\mu\) and variance \(\sigma^2\).

(a) For each \(v > 0\), the conditional distribution of \(X\) given \(V=v\) is Poisson \((v)\). Find \(E(X)\) and \(Var(X)\).

(b) For each \(v > 0\), the conditional distribution of \(X\) given \(V=v\) is gamma \((v, \lambda)\) for some fixed \(\lambda\). Find \(E(X)\) and \(Var(X)\).

**3.** The lifetime of each Type A battery has the exponential distribution with mean \(100\) hours. The lifetime of each Type B battery has the exponential distribution with mean \(150\) hours. Assume that the lifetimes of all batteries are independent of each other.

Suppose I have a packet of five batteries of which four are of Type A and one is of Type B. Let \(T\) be the lifetime of a battery picked at random from this packet. Find \(E(T)\) and \(SD(T)\).

**4.** The lifetime of each Type A battery has the exponential distribution with mean \(100\) hours. The lifetime of each Type B battery has the exponential distribution with mean \(150\) hours. Assume that the lifetimes of all batteries are independent of each other.

A factory produces large numbers of batteries, of which \(80\%\) are of Type A and \(20\%\) are of Type B. Suppose you pick batteries one by one at random from the factory’s total output until you pick a Type B battery. Let \(N\) be the number of Type A batteries that you pick, and let \(T\) be the total lifetime of these \(N\) batteries.

(a) Find \(E(N)\) and \(SD(N)\).

(b) Find \(E(T)\) and \(SD(T)\).

**5.** Think of the interval \((0, l)\) as a stick of length \(l\). The stick is broken at a point \(L_1\) chosen uniformly along it. This creates a smaller stick of random length \(L_1\) which is then broken at a point \(L_2\) chosen uniformly along it. Find \(E(L_2)\) and \(SD(L_2)\).

**6.** Let \(X_1, X_2, \ldots, X_n\) be i.i.d. with expectation \(\mu\) and variance \(\sigma^2\). Let \(S = \sum_{i=1}^n X_i\).

(a) Find the least squares predictor of \(S\) based on \(X_1\), and find the mean squared error (MSE) of the predictor.

(b) Find the least squares predictor of \(X_1\) based on \(S\), and find the MSE of the predictor. Is the predictor a linear function of \(S\)? If so, it must also be the best among all linear predictors based on \(S\), which is commonly known as the regression predictor.

[Consider whether your predictor in (b) would be different if \(X_1\) were replaced by \(X_2\), or by \(X_3\), or by \(X_i\) for any fixed \(i\). Then use symmetry and the additivity of conditional expectation.]

**7.** Let \(X\) and \(Y\) be jointly distributed random variables, and as in Section 22.1 let \(b(X) = E(Y \mid X)\). Show that \(Cov(X, Y) = Cov(X, b(X))\).

**8.** A \(p\)-coin is tossed repeatedly. Let \(W_{H}\) be the number of tosses till the first head appears, and \(W_{HH}\) the number of tosses till two consecutive heads appear.

(a) Describe a random variable \(X\) that depends only on the tosses after \(W_H\) and satisfies \(W_{HH} = W_H + X\).

(b) Use Part (a) to find \(E(W_{HH})\). What is its value when \(p = 1/2\)?

(c) Use Parts (a) and (b) to find \(Var(W_{HH})\). What is the value of \(SD(W_{HH})\) when \(p = 1/2\)?