12.5 Exercises

1. Let $X$ have the distribution given by

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

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.