The conditional expectation of a random variable \(X\) given another random variable \(Y\) is a function \(f(Y) = E[X|Y]\) defined by \(f(y) = E[X|Y=y].\)
Example: Consider rolling two 6 sided dice. Let \(X\) be the value of the first die, and let \(Y\) be the total. Find \(E[Y|X].\)
Let \(X_2\) be the value of the second die. Then \(Y = X + X_2.\) The expected value of \(X_2\) is
\[E[X_2] = \frac{1+2+3+4+5+6}{6} = 3.5\]
To compute \(E[Y|X],\) first find the value at certain points. We will start with \(E[Y|X=1].\)
\begin{align}
E[Y|X=1] & = E[X+X_2|X=1] \\
& = E[X|X=1] + E[X_2|X=1] \\
& = 1 + 3.5 \\
& = 4.5
\end{align}
At \(X = 2,\) we get
\begin{align}
E[Y|X=2] & = E[X+X_2|X=2] \\
& = E[X|X=2] + E[X_2|X=2] \\
& = 2 + 3.5 \\
& = 5.5
\end{align}
In general, if \(x \in \{1,2,3,4,5,6\},\)
\begin{align}
E[Y|X=x] & = E[X+X_2|X=x] \\
& = E[X|X=x] + E[X_2|X=x] \\
& = x + 3.5
\end{align}
So, \(E[Y|X] = X+3.5.\)
1. Let \(X_1, X_2, X_3, \dots\) be an i.i.d. sequence of random variables with Normal\((2,4)\) distributions. Let \(G\) have Geometric\((0.2)\) distribution. Define
\[Y = \sum_{i=1}^G X_i\]
Find \(E[Y|G = 3].\)
Unanswered
2. Let \(U\) be a uniform random variable on \((0, 1)\) and let \(X\) be Binomial\((12, U).\) What is \(E[X|U]?\)
Unanswered