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## 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

- \( E( R)\)
- the joint density of the min and the max of the \(U\)’s
- \(P( R>0.5)\)

[Pitman p. 355 #14]

## 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 \(\lambda\) !).

## Order statistics I

Suppose \(X_1, … , X_n \stackrel{iid}{\sim} U(0,1) \). How large must \(n\) be to have that \({\mathbf{P}}(X_{(n)} \geq .95) \geq 1/2\) ?

## Joint density part 1

Let \(X\) and \(Y\) have joint density

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

- State the marginal distribution for \(X\)
- State the marginal distribution for \(Y\)
- Are these two random variables independent?
- What is \(\mathbf{P}(Y > 2X)\)
- 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:

- the correct value of \(c\).
- \(P( X \leq x, Y \leq y)\)
- \(f_X(x)\)
- \(f_Y(y)\)
- Are \(X\) and \(Y\) independent ? Explain your reasoning ?

## Sums of normals

- Consider a normal random variable \(X\) with mean \(\mu_1\) and standard deviation \(\sigma_1\)
- Consider a normal random variable \(Y\) with mean \(\mu_2\) and standard deviation \(\sigma_2\).

Assume that \(X\) and \(Y\) are independent and define \(Z=X+Y\)

- What is the distribution of \(Z\) ?
- What is the mean and variance of \(Z\) ?
- (**) If we now assume that they are not independent, but still normal as described above, what can you say ?