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Conditional Variance
Given two random variables, we define the conditional variance of \(X\) given \(Y\) by
\[ \text{Var}(X | Y ) = E(X^2 | Y ) – (\,E(X | Y )\,)^2 \]
Show that
\[ \text{Var}(X ) = E (\text{Var}(X | Y ) ) +\text{Var}( E(X | Y ) ) \,. \]
Of course \( E(X | Y )\) is just random variable so we have that
\[ \text{Var}( E(X | Y ) ) = E [\,E(X | Y )^2\,] – [\,E(\,E(X | Y )\,)\,]^2 \]
also is is useful to recall that \(E(E(X | Y ))= E(X )\).
Random Walk
Let \(\{Z_1, Z_2, \dots, Z_n,\dots\} \) be a sequence of i.i.d random variables such that
\[ P( Z_k = a ) = \begin{cases} \frac14 & \text{ If } a=1 \\\frac14 & \text{ If } a=0\\\frac12 & \text{ If } a=-1 \end{cases}\]
If \(X_{n+1} = X_n + Z_n\) and \(X_0=1\) what is
- \( E( X_{n+1} | X_{n}) \)?
- \( E( X_{n+1}) \)?
- \( \text{Var}( X_{n+1} | X_{n}) \)?
- \( \text{Var}( X_{n+1} )\)?
Notice that \(X_n\) depends only \(Z_{n-1},Z_{n-2},\dots,Z_1\) and hence \(X_n\) is independent of \(Z_n\) !
Joint density part 2
Let \(X\) and \(Y\) have joint density
\(f(x,y) = 90(y-x)^8, \quad 0<x<y<1\)
- State the conditional distribution of \(X \mid Y\) and \(Y \mid X\)
- Are these two random variables independent?
- What is \( \mathbf{P}(Y \mid X=.2 ) \) and \( \mathbf{E}(Y \mid X=.2) \) ?
What is \( \mathbf{P}(Y \mid X=.2 ) \) and \( \mathbf{E}(Y \mid X=.2) \)
[Adapted from Pitman pg 354]
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 \(E(X)\).
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]
Expectation of hierachical model
Consider the following hierarchical random variable
- \(\lambda \sim \mbox{Geometric}(p)\)
- \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)
