(Solved): Question 3. A college collects data on its students' physical exercise time per week. Use X to deno ...

Question 3. A college collects data on its students' physical exercise time per week. Use $\mathrm{X}$ to denote a student's sex with male being 1 and female being 0 . And $\mathrm{Y}$ is the weekly physical exercise time in terms of hours. The following table shows the joint probabilities in parentheses. a) What is the marginal distribution of $Y$, and what is the mean of $Y$ ? (5 points) b) What is the male student's share in this college? What is the female student's share in this college? (5 points) c) Given $X=1$, what is the conditional distribution of $Y$ ? What is $\mathrm{E}(\mathrm{Y}\mid \mathrm{X}=1)$ ? $?(5$ points) d) Given $\mathrm{X}=0$, what is the conditional distribution of $\mathrm{Y}$ ? What is $\mathrm{E}(\mathrm{Y}\mid \mathrm{X}=0)$ ? (5 points) e) Use your answers in a) to d) to verify the Law of Iterated Expectations. ( 5 points)