# Tag Archives: JCM_math230_HW8_S13

## 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

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

[pitman p355, #14]

## 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]

## Closest Point

Consider a Poisson random scatter of points in a plane with mean intensity $$\lambda$$ per unit area. Let $$R$$ be the distance from zero to the closest point of the scatter.

1. Find a formula for the c.d.f. and the density of $$R$$ and sketch their graphs.
2. Show that $$\sqrt{2 \lambda \pi} R$$ has the Rayleigh distribution.
3. Find the mean and mode of $$R$$.

[pitman p 389, # 21]

## 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 $$(T_1,T_2,T_5)$$ ?

## Point of increase

Suppose $$U_1,U_2, …$$ are independent uniform $$(0,1)$$ random variables. Let $$N$$ be the first point of increase. That is to say the first $$n \geq 2$$ such that $$U_n > U_{n-1}$$. Show that for $$u \in (0,1)$$:

1. $\mathbf{P}(U_1 \leq u \ { and } \ N=n)= \frac{u^{n-1}}{(n-1)!}-\frac{u^{n}}{n!} \quad;\quad n \geq 2$
2. $$\mathbf{E}(N)=e$$

Some useful observations:

• $\mathbf{P}(U_1 \leq u \ { and } \ N=n) = \mathbf{P}(U_1 \leq u \ { and } \ N \geq n) -\mathbf{P}(U_1 \leq u \ { and } \ N \geq n+1)$
• The following events are equal
$\{U_1 \leq u \quad{ and } \quad N \geq n\} = \{U_{n-1}\leq U_{n-2} \leq \cdots \leq U_2\leq U_{1} \leq u \}$
• $\mathbf{P}\{U_2 \leq U_1 \leq u \}= \int_0^u \int_0^{u_1} du_2 du_1$

## 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  $$Z_1^2 + Z_2^2$$.