91Ó°ÊÓ

Given the joint pdf $$ f_{X, Y}(x, y)=2 e^{-(x+y)}, \quad 0 \leq x \leq y, \quad y \geq 0 $$ find (a) \(P(Y<1 \mid X<1)\). (b) \(P(Y<1 \mid X=1)\). (c) \(f_{Y \mid x}(y)\). (d) \(E(Y \mid x)\).

Find the conditional pdf of \(Y\) given \(x\) if $$ f_{X, Y}(x, y)=x+y $$ for \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\).

\( If $$ f_{X, Y}(x, y)=2, \quad x \geq 0, \quad y \geq 0, \quad x+y \leq 1 $$ show that the conditional pdf of \)Y\( given \)x$ is uniform.

Suppose that $$ f_{Y \mid x}(y)=\frac{2 y+4 x}{1+4 x} \quad \text { and } \quad f_{X}(x)=\frac{1}{3} \cdot(1+4 x) $$ for \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\). Find the marginal pdf for \(Y\).

Suppose that \(X\) and \(Y\) are distributed according to the joint pdf $$ f_{X, Y}(x, y)=\frac{2}{5} \cdot(2 x+3 y), \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1 $$ Find (a) \(f_{X}(x)\). (b) \(f_{Y \mid x}(y)\). (c) \(P\left(\frac{1}{4} \leq Y \leq \frac{3}{4} \mid X=\frac{1}{2}\right)\). (d) \(E(Y \mid x)\).

If \(X\) and \(Y\) have the joint pdf $$ f_{X, Y}(x, y)=2, \quad 0 \leq x

Find \(P\left(X<1 \mid Y=1 \frac{1}{2}\right)\) if \(X\) and \(Y\) have the joint pdf $$ f_{X, Y}(x, y)=x y / 2, \quad 0 \leq x

Suppose that \(X_{1}, X_{2}, X_{3}, X_{4}\), and \(X_{5}\) have the joint pdf $$ f x_{1}, X_{2}, X_{3}, X_{4}, X_{5}\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=32 x_{1} x_{2} x_{3} x_{4} x_{5} $$ for \(0 \leq x_{i} \leq 1, i=1,2, \ldots, 5\). Find the joint conditional pdf of \(X_{1}, \bar{X}_{2}\), and \(X_{3}\) given that \(X_{4}=x_{4}\) and \(X_{5}=x_{5}\).

Suppose the random variables \(X\) and \(Y\) are jointly distributed according to the pdf $$ f_{X, Y}(x, y)=\frac{6}{7}\left(x^{2}+\frac{x y}{2}\right), \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2 $$ Find (a) \(f_{X}(x)\). (b) \(P(X>2 Y)\). (c) \(P\left(Y>1 \mid X>\frac{1}{2}\right)\).

For continuous random variables \(X\) and \(Y\), prove that \(E(Y)=E_{X}[E(Y \mid x)]\).