# Tag Archives: JCM_math340_HW8_F13

## Joint, Marginal and Conditioning

Let $$(X,Y)$$ have joint density $$f(x,y) = e^{-y}$$, for $$0<x<y$$, and $$f(x,y)=0$$ elsewhere.

1. Are $$X$$ and $$Y$$ independent ?
2. Compute the marginal density of $$Y$$.
3. Show that $$f_{X|Y}(x,y)=\frac1y$$, for $$0<x<y$$.
4. Compute $$E(X|Y=y)$$
5. Use the previous result to find $$E(X)$$.

## Three Random Variables

Let $$X$$, $$Y$$, and $$Z$$ be independent uniform $$(0,1)$$.

1. Find the joint density of  $$XY$$ and $$Z^2$$.
2. Show that $$P(XY < Z^2) = \frac59$$.

[Meester ex. 5.12.25]

## A joint density example I

Let $$(X,Y)$$ have joint density $$f(x,y)=x e^{-x-y}$$ when $$x,y>0$$ and $$f(x,y)=0$$ elsewhere. Are $$X$$ and $$Y$$ independent ?

[Meester ex 5.12.30]

## Memory and the Exponential

Let $$X$$ have an exponential distribution with parameter $$\lambda$$. Show that

$P( X> t+ s \,|\, X>s) = P(X>t)$

for all $$s,t >0$$. Explain why one might call this property of the exponential “the lack of memory”.

## conditional densities

Let $$X$$ and $$Y$$ have the following joint density:

$f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\ 0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}$

1. Find the marginal densities of $$X$$ and $$Y$$
2. find $$f_{Y|X}( y \,|\, X=\frac14)$$
3. find $$\mathbf{E}(Y \,|\, X=\frac14)$$

[Pitman p426 # 2]

## A Joint density example II

If $$X$$ and $$Y$$ have joint density function

$f(x,y)=\frac{1}{x^2y^2} \quad; \quad x \geq 1, y\geq 1$

1.  Compute the joint density fiction of  $$U=XY$$, $$V=X/Y$$.
2. What are the marginal densities of $$U$$ and $$V$$ ?

[Ross p295, # 54]

## An example of min and change of variable

Suppose $$R_1$$ and $$R_2$$ are two independent random variables with the same density function

$f(x)=x\exp(-{\textstyle \frac12 }x^2)$

for $$x\geq 0$$. Find

1. the density of $$Y=\min(R_1,R_2)$$;
2. the density of $$Y^2$$

[Pitman p. 336 #21]

## Box-Muller I

Let $$U_1$$ and $$U_2$$ be independent random variables distributed uniformly on $$(0,1)$$.

define $$(Z_1,Z_2)$$ by

$Z_1=\sqrt{ -2 \log(U_1) }\cos( 2 \pi U_2)$

$Z_2=\sqrt{ -2 \log(U_1) }\sin( 2 \pi U_2)$

1. Find the joint density of $$(Z_1, Z_2)$$.
2. Are $$Z_1$$ and $$Z_2$$ independent ? why ?
3. What is the marginal density of $$Z_1$$ and $$Z_2$$ ? Do you recognize it ?
4. Reflect on the implications of the previous answer for generating an often needed class of random variable on a computer.

Hint: To eliminate $$U_1$$ write the formula for  $$Z_1^2 + Z_2^2$$.