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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.