# 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.