3.20 Extra Questions

  1. Suppose that \(\E[X] = 10\) and \(\var(X) = 2\). Also, suppose that \(Y=5 + 9 X\).

    1. What is \(\E[Y]\)?

    2. What is \(\var(Y)\)?

  2. Use the definition of variance to show that \(\Var(bX) = b^2 \Var(X)\) (where \(b\) is a constant and \(X\) is some random variable).

  3. Suppose you are interested in average height of students at UGA. Let \(Y\) denote a student’s height; also let \(X\) denote a binary variable that is equal to 1 if a student is female. Suppose that you know that \(\E[Y|X=1] = 5' \,4''\) and that \(\E[Y|X=0] = 5' \,9''\) (and that \(\P(X=1) = 0.5\)).

    1. What is \(\E[Y]\)?

    2. Explain how the answer to part (a) is related to the Law of Iterated Expectations.

  4. Consider a random variable \(X\) with support \(\mathcal{X} = \{2,7,13,21\}\). Suppose that it has the following pmf:

    \[ \begin{aligned} f_X(2) &= 0.5 \\ f_X(7) &= 0.25 \\ f_X(13) &= 0.15 \\ f_X(21) &= ?? \end{aligned} \]

    1. What is \(f_X(21)\)? How do you know?

    2. What is the expected value of \(X\)? [Show your calculation.]

    3. What is the variance of \(X\)? [Show your calculation.]

    4. Calculate \(F_X(x)\) for \(x=1\), \(x=7\), \(x=8\), and \(x=25\).