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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 \(p = 1/2\)?