# Category Archives: Joint Distributions

## Joint Density Poisson arrival

Let $$T_1$$ and $$T_5$$ be the times of the first and fifth arrivals in a Poisson arrival prices with rate $$\lambda$$. Find the joint distribution of $$T_1$$ and $$T_5$$ .

## Uniform Spacing

Let $$U_1, U_2, U_3, U_4, U,5$$ be independent uniform $$(0,1)$$ random variables. Let $$R$$ be the difference between the max and the min of the random variables. Find

1. $$E( R)$$
2. the joint density of the min and the max of the $$U$$’s
3. $$P( R>0.5)$$

[Pitman p. 355 #14]

## Joint density part 2

Let $$X$$ and $$Y$$ have joint density

$$f(x,y) = 90(y-x)^8, \quad 0<x<y<1$$

1. State the conditional distribution of $$X \mid Y$$ and $$Y \mid X$$
2. Are these two random variables independent?
3. What is $$\mathbf{P}(Y \mid X=.2 )$$ and $$\mathbf{E}(Y \mid X=.2)$$ ?

What is $$\mathbf{P}(Y \mid X=.2 )$$ and $$\mathbf{E}(Y \mid X=.2)$$

[Adapted from Pitman pg 354]

## Uniform distributed points given an arrival

Consider a Poisson arrival process with rate $$\lambda>0$$. Let $$T$$ be the time of the first arrival starting from time $$t>0$$. Let $$N(s,t]$$ be the number of arrivals in the time interval $$(s,t]$$.

Fixing an $$L>0$$, define the pdf $$f(t)$$ by $$f(t)dt= P(T \in dt | N(0,L]=1)$$ for $$t \in (0,L]$$. Show that $$f(t)$$ is the pdf of a uniform random variable on the interval $$[0,L]$$ (independent of $$\lambda$$ !).

## Joint Distribution Table

Consider the following joint distribution.

If the experiment is flipping a fair coin three times, which of the following could be the random variables $$X$$ and $$Y$$. Select all that apply.

1. $$X=$$ the number of heads, $$Y=$$ the number of tails.
2. $$X=$$ the number of tails, $$Y=$$ the number of tails (i.e., $$Y=X$$).
3. $$X=$$ the number of heads. $$Y=3-X.$$
4. $$X=$$ the number of tails on the first two flips. $$Y=$$ the number of tails on the last two flips.

## 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 $$E(X)$$.

## 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]

## 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]

## Joint of min and max

Let $$X_1,…,X_n \stackrel{iid}{\sim} \mbox{Exp}(\lambda)$$

Let $$V = \mbox{min}(X_1,…,X_n)$$ and  $$W = \mbox{max}(X_1,…,X_n)$$.

What is the joint distribution of $$V,W$$. Are they independent ?

## Joint density part 1

Let $$X$$ and $$Y$$ have joint density

$$f(x,y) = 90(y-x)^8, \quad 0<x<y<1$$

1. State the marginal distribution for $$X$$
2. State the marginal distribution for $$Y$$
3. Are these two random variables independent?
4. What is $$\mathbf{P}(Y > 2X)$$
5. Fill in the blanks “The density $$f(x,y)$$ above   is the joint density of the  _________ and __________ of ten independent uniform $$(0,1)$$ random variables.”

[Adapted from Pitman pg 354]

## Simple Joint density

Let $$X$$ and $$Y$$ have joint density

$f(x,y) = c e^{-2x -3 y} \quad (x,y>0)$

for some $$c>0$$ and $$f(x,y)=0$$ otherwise. find:

1. the correct value of $$c$$.
2. $$P( X \leq x, Y \leq y)$$
3. $$f_X(x)$$
4. $$f_Y(y)$$
5. Are $$X$$ and $$Y$$ independent ? Explain your reasoning ?

## Joint Distributions of Uniforms

Let $$X$$ and $$Y$$ be independent, each uniformly distributed on $$\{1,2,\dots,n\}$$. Find:

1. $$\mathbf{P}( X=Y)$$
2. $$\mathbf{P}( X < Y)$$
3. $$\mathbf{P}( X>Y)$$
4. $$\mathbf{P}( \text{max}(X,Y)=k )$$ for $$k=1,\dots,n$$
5. $$\mathbf{P}( \text{min}(X,Y)=k )$$ for $$k=1,\dots,n$$
6. $$\mathbf{P}( X+Y=k )$$ for $$k=2,\dots,2n$$