Discrete Random Variables (DRV)

IB AA SL: core and optional content

Core: expectation. Optional: general discrete-random-variable variance and standard deviation.

Expectation of a Discrete Random Variable

The expectation (mean) of a discreate random variable \(X\) indicates its weighted average and its give by

\(\mathrm{E}\left( X \right)=\sum\limits_{\text{all}\hspace{0.5em}x}{x\mathrm{P}\left( X=x \right)}\text{ or}\hspace{0.5em}\overline{x}=\frac{\sum{fx}}{\sum{f}}\)

Variance of a Discrete Random Variable

\(\mathrm{Var}\left( X \right)\) is the measure of the expected spread of a random variable about its mean and is a nonnegative number. It is zero when the random variable is constant (with probability 1).

\(\mathrm{Var}\left( X \right)=\mathrm{E}\left[ {{\left( X-\mu \right)}^{2}} \right]=\mathrm{E}\left( {{X}^{2}} \right)-{{\left[ \mathrm{E}\left( X \right) \right]}^{2}}\)
Standard deviation, \(\sigma =\sqrt{\mathrm{Var}\left( X \right)}\)