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  \(Z_1^2 + Z_2^2\).