5 Expectation and Variability Free Online FlashCards

What does expectation describe?

Expectation is the probability-weighted average of a random variable. It describes the long-run average or center of its distribution.

Does expectation require independence to add?

Expectation is linear: $E[aX+bY]=aE[X]+bE[Y]$. This holds whether or not $X$ and $Y$ are independent.

How is the expectation of a discrete variable calculated?

For a discrete variable,
$E[X]=\sum_x x p_X(x)$. For a continuous variable,
$E[X]=\int_{-\infty}^{\infty}x f_X(x)\,dx$.

What is the computational formula for variance?

For $\mu=E[X]$, variance is $\operatorname{Var}(X)=E[(X-\mu)^2]$. Equivalently, $\operatorname{Var}(X)=E[X^2]-(E[X])^2$.

When is a random variable's variance zero?

Variance is always nonnegative and equals zero exactly when the random variable is constant with probability one.

How does an affine transformation affect variance?

$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$. Adding $b$ does not change spread; scaling by $a$ scales variance by $a^2$.

What is standard deviation?

Standard deviation is the positive square root of variance: $\operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}$. It has the same units as $X$.

What does covariance measure?

Covariance measures whether two variables tend to be above or below their means together: $\operatorname{Cov}(X,Y)=E[XY]-E[X]E[Y]$.

Does zero covariance imply independence?

Independence implies $\operatorname{Cov}(X,Y)=0$, but zero covariance generally does not imply independence. Zero covariance rules out linear association, not all dependence.

What is the variance of a sum?

$\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y)$. The covariance term accounts for joint movement.

What adds for independent random variables?

For independent variables, $\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$. Standard deviations generally do not add.

How does the standard deviation of an independent sum grow?

If independent variables share variance $\sigma^2$, then for $S=X_1+\cdots+X_n$, $\operatorname{Var}(S)=n\sigma^2$ and $\operatorname{SD}(S)=\sigma\sqrt{n}$.