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## 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.

- Are \(X\) and \(Y\) independent ?
- Compute the marginal density of \(Y\).
- Show that \(f_{X|Y}(x,y)=\frac1y \), for \(0<x<y\).
- Compute \(E(X|Y=y)\)
- 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]

## conditional densities

Let \(X\) and \(Y\) have the following joint density:

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

- Find the marginal densities of \(X\) and \(Y\)
- find \(f_{Y|X}( y \,|\, X=\frac14)\)
- find \( \mathbf{E}(Y \,|\, X=\frac14)\)

[Pitman p426 # 2]

## Difference between max and min

Let \(U_1,U_2,U_3,U_4,U_5\) be independent, each with uiform distribution on \((0,1)\). Let \(R\) be the distance between the max and the min of the \(U_i\)’s. Find

- \(\mathbf{E} R\)
- the joint density of the max and the min of the \(U_i\)’s.
- the \(\mathbf{P}(R> .5)\)

[pitman p355, #14]

## Joint Density of Arrival Times

Let \(T_1 < T_2<\cdots\) be the arrival times in a Poisson arrival process with rate \(\lambda\). What is the joint distribution of \((T_1,T_2,T_5)\) ?

## Conditioning and Polya’s urn

An urn contains 1 black and 2 white balls. One ball is drawn at random and its color noted. The ball is replaced in the urn, together with an additional ball of its color. There are now four balls in the urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way.

- Let \(B_n\) be the number of black balls in the urn just before the \(n\)th ball is drawn. (Thus \(B_1= 1\).) For \(n \geq 1\), find \(\mathbf{E} (B_{n+1} | B_{n}) \).
- For \(n \geq 1\), find \(\mathbf{E} (B_{n}) \). [Hint: Use induction based on the previous answer and the fact that \(\mathbf{E}(B_1) =1\)]
- For \(n \geq 1\), what is the expected proportion of black balls in the urn just before the \(n\)th ball is drawn ?

[From pitman p 408, #6]

## Expectation of hierachical model

Consider the following hierarchical random variable

- \(\lambda \sim \mbox{Geometric}(p)\)
- \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)