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17.5. Exercises

1. A Prob 140 student comes to lecture at a time that is uniformly distributed between 5:09 and 5:14. Independently of the student, the professor begins the lecture at a time that is uniformly distributed between 5:10 and 5:12. What is the chance that the lecture has already begun when the student arrives?

2. For some constant \(c\) let \(X\) and \(Y\) have a joint density given by

\[\begin{split}f(x, y) ~ = ~ \begin{cases} c(x - y), ~~ 0 < y < x < 1 \\ 0 ~~~~~~~~~~~~~~ \text{otherwise} \end{cases}\end{split}\]

(a) Draw the region over which \(f\) is positive.

(b) Find \(c\).

(c) Find \(P(X > Y + 0.4)\). Before you calculate, shade the event on the diagram you drew in (a).

(d) Find the density of \(X\).

(e) Are \(X\) and \(Y\) independent?

(f) Find \(E(XY)\).

3. Let \(X\) and \(Y\) have joint density \(f\) given by

\[\begin{split}f(x, y) ~ = ~ \begin{cases} \frac{40}{243}x(3-x)(x-y), ~~~~ 0 < y < x < 3 \\ 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{otherwise} \end{cases}\end{split}\]

Write the each of the following in terms of \(f\) but do not simplify the expression. Integrals should not include regions where \(f(x, y) = 0\).

(a) \(P(Y > 1)\)

(b) the conditional density of \(X\) given \(Y = 1\) (please be clear about the values on which the density is positive)

(c) \(E(e^{XY})\)

4. The joint density of \(X\) and \(Y\) is

\[\begin{split}f(x, y) ~ = ~ \begin{cases} 24xy, ~~~ x, y > 0 \text{ and } 0 < x+y < 1 \\ 0 ~~~~~~~~~~~~~ \text{otherwise} \end{cases}\end{split}\]

(a) Find the density of \(X\). Recognize this as one of the famous ones and state its name and parameters.

(b) Without further calculation, find the density of \(Y\) and justify your answer.

(c) Are \(X\) and \(Y\) independent? Why or why not?

(d) Find the conditional density of \(X\) given \(Y = 0.75\). As always, start with the possible values.

(e) Find \(P(X > 0.2 \mid Y = 0.75)\).

(f) Find \(E(X \mid Y = 0.75)\).

5. Let \(U_i\), \(1 \le i \le 20\) be i.i.d. uniform \((0, 1)\) variables, and let \(U_{(k)}\) be the \(k\)th order statistic.

(a) What is the density of \(U_{(7)}\)?

(b) Without integrating the density, find the cdf of \(U_{(7)}\).

[Draw a line representing the unit interval and put down crosses representing the variables. For \(U_{(7)}\) to be less than \(x\), how must you distribute the crosses?]

(c) Find the joint density of \(U_{(7)}\) and \(U_{(12)}\).

6. A random variable \(X\) has the beta \((2, 2)\) density. Given \(X = x\), the conditional distribution of the random variable \(Y\) is uniform on the interval \((-x, x)\).

(a) Find \(P(Y < 0.2 \mid X = 0.6)\).

(b) Find \(E(Y)\).

(c) Find the joint density of \(X\) and \(Y\). Remember to specify the region where it is positive.

(d) Find \(P(X < 0.3, \vert Y \vert < 0.3X)\).

7. Let \(X\) and \(Y\) be i.i.d. with a joint density.

(a) Find \(P(Y > X)\).

(b) Find \(P(\vert Y \vert > \vert X \vert)\).

(c) If \(X\) and \(Y\) are i.i.d. standard normal, find \(P(Y > \vert X \vert)\).

8. Two points are placed independently and uniformly at random on the unit interval. This creates three segments of the interval. What is the chance that the three segments can form a triangle?

[To find probabilities of events determined by two independent uniform \((0,1)\) random variables, it’s a good idea to draw the unit square.]

9. Let \(X\) have the beta \((r, s)\) density. Find \(Var(X)\). You can assume \(r\) and \(s\) are integers.