Conditional Expectation
- \(\displaystyle\operatorname{E}(X\vert Y = y) = \sum_{x_i} x_ip_{X\vert Y}(x_i \vert y) = \sum_{x_i} x_i\frac{p_{X, Y}(x_i, y)}{p_Y(y)}\)
- \(\displaystyle\operatorname{E}(X\vert Y = y) = \int_{-\infty}^\infty x f_{X\vert Y}(x \vert y)\;dx = \int_{-\infty}^\infty x \frac{f_{X, Y}(x, y)}{f_Y(y)}\;dx\)
- \[\operatorname{E}\left(\operatorname{E}\left(X\vert Y\right)\right) = \operatorname{E}X\]
- \(\displaystyle\operatorname{E}X = \sum_{y_i} \operatorname{E}(X\vert Y = y_i)\operatorname{P}(Y = y_i) = \sum_{y_i} \operatorname{E}(X\vert Y = y_i)p_Y(y_i)\)
- \(\displaystyle\operatorname{E}X = \int_{-\infty}^\infty \operatorname{E}(X\vert Y = y)f_Y(y)\;dy\)
Expected Value of Geometric Distribution
Probability using Conditioning
Given an event \(A\), let \(X\) be the indicator variable of \(A\):
\[X = \begin{cases}1 & \text{ if $A$ occurs}\\0 & \text{
otherwise}\end{cases}\]
\(X \sim \operatorname{Bernoulli}(\operatorname{P}(A))\)
\(\operatorname{E}X = \operatorname{P}(A)\)
For any random variable \(Y\), \(\operatorname{E}(X\vert Y = y) =
\operatorname{P}(A\vert Y = y)\)
Then
- \(\displaystyle\operatorname{P}(A) = \operatorname{E}X = \sum_{y_i} \operatorname{E}(X\vert Y = y_i)\operatorname{P}(Y = y_i) = \sum_{y_i} \operatorname{P}(A\vert Y = y_i)p_Y(y_i)\)
- \(\displaystyle\operatorname{P}(A) = \operatorname{E}X = \int_{-\infty}^\infty \operatorname{E}(X\vert Y = y)f_Y(y)\;dy = \int_{-\infty}^\infty \operatorname{P}(A\vert Y = y)f_Y(y)\;dy\)
Example 1
Let \(X\) and \(Y\) be two independent continuous random variables. Calculate the probability \(\operatorname{P}(X < Y)\).
Example 2
Let \(X_1, X_2, \ldots, X_n\) be independent random variables, \(X_i \sim
\operatorname{Bernoulli(p_i)}\).
Let \(Y = X_1 + X_2 + \cdots + X_n\).
Find the probability mass function of \(Y\).