# Tag Archives: JCM_math230_HW11_F22

## 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

1.  $$E( X_{n+1} | X_{n})$$?
2.  $$E( X_{n+1})$$?
3.  $$\text{Var}( X_{n+1} | X_{n})$$?
4. $$\text{Var}( X_{n+1} )$$?

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

$$f(x,y) = 90(y-x)^8, \quad 0<x<y<1$$

1. State the conditional distribution of $$X \mid Y$$ and $$Y \mid X$$
2. Are these two random variables independent?
3. What is $$\mathbf{P}(Y \mid X=.2 )$$ and $$\mathbf{E}(Y \mid X=.2)$$ ?

What is $$\mathbf{P}(Y \mid X=.2 )$$ and $$\mathbf{E}(Y \mid X=.2)$$

## 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 $$E(X)$$.

## 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]

## Expectation of hierachical model

Consider the following hierarchical random variable

1. $$\lambda \sim \mbox{Geometric}(p)$$
2. $$Y \mid \lambda \sim \mbox{Poisson}(\lambda)$$
Compute $$\mathbf{E}(Y)$$.