## 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 $$x=25$$.