Practice set 5.3

1

Suppose \(X\) and \(Y\) are jointly continuous with joint pdf given by:

\[ f(x, y) = \begin{cases} 3x, & 0 \le y \le x \le 1\\ 0, & \text{elsewhere.} \end{cases} \]

A

Find the marginal density functions of \(X\) and \(Y\). Do either of them follow well-known forms?

B

For what values of \(x\) is \(f_{Y|X}(y|x)\) defined? Find this conditional density, and identify it as following a well-known form.

C

Find \(F_{Y|X}(y|x)\).

D

What is \(P(Y>1/2 | X = 3/4)\)?

E

What is \(P(Y<1/2 | X = 1/4)\)?

2

Let X and Y have joint density function:

\[f_{X,Y}(x, y) = \begin{cases} e^{-(x + y)}, & x > 0, \; y > 0, \\ 0, & \text{elsewhere.} \end{cases}\]

with joint CDF:

\[F_{X,Y}(x,y) = \begin{cases} 0, & x \le 0 \text{ or } y \le 0, \\ (1 - e^{-x})(1 - e^{-y}), & x > 0, y > 0 \end{cases}\]

Find the marginal CDFs of \(X\) and \(Y\). Identify \(X\) and \(Y\) as marginally following well-known forms.

3

\(X\) and \(Y\) are jointly continuous with joint pdf:

\[f_{X,Y}(x,y) = \begin{cases}6(1-y) & 0 < x < y < 1 \\ 0 & otherwise \end{cases}\]

and joint CDF:

\[F_{X,Y}(x,y) = \begin{cases} 0 & x < 0\ or\ y < 0 \\ 3x^2-2x^3 + 3x(y-x)(2-y-x) & 0 < x < y < 1 \\ 3y^2-2y^3 & 0 < y < 1, x \geq y \\ x^3-3x^2+3x & 0 < x < 1, y\geq 1\\ 1 & x\geq 1, y \geq 1 \end{cases}\]

A

Find \(F_{Y|X}(y|x)\).

B

Find \(F_{X|Y}(x|y)\).

4

Suppose that the random variables X and Y have joint probability density function given by

\[f(x, y) = \begin{cases} 6x^2y, & 0 \le x \le y,\; x + y \le 2, \\ 0, & \text{elsewhere.} \end{cases}\]

A

Find the marginal pdfs of \(X\) and \(Y\). Do either of them follow well-known forms?

B

For what values of \(x\) does \(f_{Y|X}(y|x)\) exist? Find this conditional density.

C

Find \(F_{Y|X}(y|x)\).

D

Find \(P(Y<1.1 | X = 0.6)\).

E

Find \(P(Y>1.5 | X = 0.8)\).

F

Find \(P(Y > 1.5 | X = 1.6)\).

5

Suppose \(X\) and \(Y\) are uniformly distributed over the triangular region shown below:

A

Find the marginal density functions.

B

What is \(P(Y>1/2 | X = 1/4)\)?

6

If \(X\sim UNIF(0,1)\) and for \(0<x<1\), \(Y|X=x \sim UNIF(0,x)\):

A

Find the joint pdf of \(X\) and \(Y\).

B

Find the marginal density function for \(Y\).

7

If \(X\sim N(0,1)\) and \(Y|X=x \sim N(x, 1)\), show that marginally \(Y\sim N(0,2)\).