# 12.5. Exercises¶

1. Let $$X$$ have the distribution given by

$$~~~~~~~~~~~~~~~~~~~~~~~~~x$$

1

2

3

$$P(X=x)$$

0.2

0.5

0.3

Find $$E(X)$$ and $$SD(X)$$.

2. Suppose $$P(X = x_0) = 1-p$$ and $$P(X = x_1) = p$$ for values $$x_0$$ and $$x_1$$ such that $$x_1 > x_0$$.

Write $$X$$ as a linear function of an indicator that has value $$1$$ with probability $$p$$, and hence find $$SD(X)$$ in terms of $$p$$, $$q=1-p$$, and $$d=x_1-x_0$$.

3. Let $$X$$ have the Poisson $$(\mu)$$ distribution and let $$Y$$ have the Poisson $$(\lambda)$$ distribution independent of $$X$$.

(a) What is the distribution of $$X+Y$$?

(b) Which of the following statements is (or are) true? Pick all that are true and justify your choices.

(i) $$E(X+Y) = E(X) + E(Y)$$

(ii) $$SD(X+Y) = SD(X) + SD(Y)$$

(iii) $$Var(X+Y) = Var(X) + Var(Y)$$

4. Let $$p∈(0,1)$$ and let $$X$$ be the number of spots showing on a flattened die that shows its six faces according to the following chances:

$$P(X=1)=P(X=6)$$

$$P(X=2)=P(X=3)=P(X=4)=P(X=5)$$

$$P(X=1$$ or $$6)=p$$

Find $$SD(X)$$ and explain why it is an increasing function of $$p$$. Compare your answer with the answer you got for the mean absolute deviation in Chapter 8.

5. Consider a sequence of i.i.d. Bernoulli $$(p)$$ trials. Let $$T$$ be the number of trials till the first success and let $$F$$ be the number of failures before the first success. You know that $$T$$ has the geometric $$(p)$$ distribution on $$\{1, 2, 3, \ldots\}$$ and that $$E(T) = \frac{1}{p}$$. We will show later, by conditioning, that $$SD(T) = \frac{\sqrt{q}}{p}$$ where $$q = 1-p$$. For now you can just assume that it is true.

(a) Write $$F$$ as a function of $$T$$ and hence find $$E(F)$$ and $$SD(F)$$.

(b) Find the distribution of $$F$$. It is called the geometric $$(p)$$ distribution on $$\{0, 1, 2, \ldots \}$$.

6. A random variable $$X$$ has expectation $$20$$ and SD $$2$$. Find the best upper and lower bounds you can on

(a) $$P(15 < X < 25)$$

(b) $$P(15 < X < 30)$$

7. Consider a probabilistic model that has a numerical parameter $$\theta$$. A “probabilistic model” is just a set of assumptions about randomness. Let the random variable $$T$$ be an estimator of $$\theta$$. Frequently, $$T$$ is a statistic based on a random sample.

Recall that the bias of $$T$$ is defined as $$B_\theta (T) ~ = ~ E_\theta (T) - \theta$$, where the subscript $$\theta$$ reminds us that $$\theta$$ is the true value of the parameter.

The mean squared error of the estimator $$T$$ is $$MSE_\theta (T) ~ = ~ E_\theta \big{(} (T - \theta)^2 \big{)}$$.

Follow the calculation in Section 12.2 of the textbook to show the bias-variance decomposition given by

$MSE_\theta (T) ~ = ~ Var_\theta(T) + B_\theta^2(T)$

Note that the square in the bias term makes sense. Bias has the same units as $$T$$, whereas the MSE and variance are in the square of those units.