Chapter 5.4, Problem 58E

Suppose that the random variables Y₁ and Y₂ have joint probability density function, f(y₁, y₂), given by (see Exercise 5.14)

$f\left(y_1,y_2\right)=\{\begin{array}{l}6y_1^2{y}_2,0\le y_1\le y_2,y_1+y_2\le 2,\\ 0,\text{elsewhere}\text{.}\end{array}$

Show that Y₁ and Y₂ are dependent random variables.

5.14 Suppose that the random variables Y₁ and Y₂ have joint probability density function f(y₁, y₂) given by

$f\left(y_1,y_2\right)=\{\begin{array}{l}6y_1^2{y}_2,0\le y_1\le y_2,y_1+y_2\le 2,\\ 0,\text{elsewhere}\text{.}\end{array}$

1. a Verify that this is a valid joint density function.
2. b What is the probability that Y₁ + Y₂ is less than 1?
3. 5.32. Suppose that the random variables Y₁ and Y₂ have joint probability density function, f(y₁, y₂), given by (see Exercise 5.14)

   $f\left(y_1,y_2\right)=\{\begin{array}{l}6y_1^2{y}_2,0\le y_1\le y_2,y_1+y_2\le 2,\\ 0,\text{elsewhere}\text{.}\end{array}$

4. a Show that the marginal density of Y₁ is a beta density with α = 3 and β = 2.
5. b Derive the marginal density of Y₂.
6. c Derive the conditional density of Y₂ given Y₁ = y₁.
7. d Find P(Y₂ < 1.1|Y₁ =.60).