Tag Archives: JCM_math340_HW8_F13

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.

  1. Are \(X\) and \(Y\) independent ?
  2. Compute the marginal density of \(Y\).
  3. Show that \(f_{X|Y}(x,y)=\frac1y \), for \(0<x<y\).
  4. Compute \(E(X|Y=y)\)
  5. Use the previous result to find \(E(X)\).

Three Random Variables

Let \(X\), \(Y\), and \(Z\) be independent uniform \( (0,1)\).

  1. Find the joint density of  \(XY\) and \(Z^2\).
  2. Show that \(P(XY < Z^2) = \frac59\).

 

[Meester ex. 5.12.25]

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]

Memory and the Exponential

Let \(X\) have an exponential distribution with parameter \(\lambda\). Show that

\[ P( X> t+ s \,|\, X>s) = P(X>t) \]

for all \(s,t >0\). Explain why one might call this property of the exponential “the lack of memory”.

conditional densities

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

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

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

[Pitman p426 # 2]

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]

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  \(Z_1^2 + Z_2^2\).