| |
\[
\begin{aligned}
var(X) &= \mathbb{E}\left[(X - \mathbb{E}[X])^2\right] \\
&= \mathbb{E}\left[X^2\right] - \big(\mathbb{E}[X]\big)^2 \\
var(X\,|\,Y=y) &= \mathbb{E}\left[\left(X - \mathbb{E}[X\,|\,Y=y]\right)^2\,|\,Y=y\right] \\
\end{aligned}
\] |
Law of total variance
| |
\[
\begin{aligned}
var(X) &= \mathbb{E}\left[var(X\,|\,Y)\right] + var\big(\mathbb{E}[X\,|\,Y]\big) \\
var(X_1 + \cdots + X_N) &= \mathbb{E}\left[var(X_1 + \cdots + X_N\,|\,N)\right] + var\big(\mathbb{E}[X_1 + \cdots + X_N \,|\,N]\big) \\
&= \mathbb{E}[N] \cdot var(X) + \big(\mathbb{E}[X]\big)^2\cdot var(N) \\
\end{aligned}
\] |
Covaraince
| |
\[
\begin{aligned}
cov(X,Y) &= \mathbb{E}\left[(X - \mathbb{E}[X])\cdot(Y - \mathbb{E}[Y])\right] \\
&= \mathbb{E}\left[XY\right] - \mathbb{E}[X]\cdot \mathbb{E}[Y] \\
cov(aX+b,Y) &= a\cdot cov(X,Y) \\
cov(X,Y+Z) &= cov(X,Y) + cov(X,Z) \\
\end{aligned}
\] |
Correlation coefficient
| |
\[
\begin{aligned}
\rho(X,Y) &= \frac{cov(X,Y)}{\sigma_X \sigma_Y} & -1 \le \rho \le 1\\
\end{aligned}
\] |
Sum of random variables
| |
\[
\begin{aligned}
var(X_1 + X_2) &= var(X_1) + var(X_2) + 2cov(X_1,X_2) \\
var(X_1 - X_2) &= var(X_1) + var(X_2) - 2cov(X_1,X_2) \\
\end{aligned}
\] |
Source: MITx 6.041x, Lecture 12, 13.
# posted by rot13(Unafba Pune) @ 8:55 PM
