# Category Archives: Conditional Variance

## 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 $$Z_n$$ !