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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.
- Are \(X\) and \(Y\) independent ?
- Compute the marginal density of \(Y\).
- Show that \(f_{X|Y}(x,y)=\frac1y \), for \(0<x<y\).
- Compute \(E(X|Y=y)\)
- Use the previous result to find \(E(X)\).
Three Random Variables
Let \(X\), \(Y\), and \(Z\) be independent uniform \( (0,1)\).
- Find the joint density of \(XY\) and \(Z^2\).
- 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:
\[ f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\ 0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}\]
- Find the marginal densities of \(X\) and \(Y\)
- find \(f_{Y|X}( y \,|\, X=\frac14)\)
- find \( \mathbf{E}(Y \,|\, X=\frac14)\)
[Pitman p426 # 2]
A Joint density example II
If \(X\) and \(Y\) have joint density function
\[f(x,y)=\frac{1}{x^2y^2} \quad; \quad x \geq 1, y\geq 1\]
- Compute the joint density fiction of \(U=XY\), \(V=X/Y\).
- 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
- the density of \(Y=\min(R_1,R_2)\);
- 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
\[Z_1=\sqrt{ -2 \log(U_1) }\cos( 2 \pi U_2) \]
\[Z_2=\sqrt{ -2 \log(U_1) }\sin( 2 \pi U_2) \]
- Find the joint density of \((Z_1, Z_2)\).
- Are \(Z_1\) and \(Z_2\) independent ? why ?
- What is the marginal density of \(Z_1\) and \(Z_2\) ? Do you recognize it ?
- 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 \(Z_1^2 + Z_2^2\).
