# Tag Archives: JCM_math230_HW10_F22

## 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]

## 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

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

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.”

## 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 ?

## 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 $$Z=X+Y$$

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