Probability MT22, Variance and covariance
Flashcards
@State the two equivalent definitions of $\mathrm{Var}(X)$.
\[\mathrm{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\]
How can you interpret the variance of a random variable?
The average squared distance from the mean.
Given that $X$ and $Y$ are independent, what is the formula for $\mathbb{E}[XY]$?
\[\mathbb{E}[XY] = \mathbb{E}[X]\mathbb{E}[Y]\]
@State the two equivalent definitions for $\mathrm{Cov}(X, Y)$.
\[\mathrm{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y-\mathbb{E}[Y])] = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]\]
Given that $X$ and $Y$ are independent, what is $\mathrm{Cov}(X, Y)$?
\[0\]
If $X$ and $Y$ are independent, then $\mathrm{Cov}(X, Y) = 0$. Is the converse true?
No.
What is $\mathrm{Var}(X+Y)$?
\[\mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\mathrm{Cov}(X, Y)\]