# 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 $$p = 1/2$$?