Theory

Theory 1

Expected value

The expected value $E[X]$ of random variable $X$ is the weighted average of the values of $X$, weighted by the probability of those values.

Discrete formula using PMF:

$$E[X]=\sum _{k}k\cdot P_X(k)$$

Continuous formula using PDF:

$$E[X]={\int }_{-\infty }^{+\infty }x\cdot f_X(x)dx$$

Notes:

• Expected value is sometimes called expectation, or even just mean, although the latter is best reserved for statistics.
• The Greek letter $\mu$ is also used in contexts where ‘mean’ is used.

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Let $X$ be a random variable, and write $E[X]=\mu$.

Variance

The variance $\mathrm{Var}[X]$ measures the average squared deviation of $X$ from $\mu$. It estimates how concentrated $X$ is around $\mu$.

• Defining formula:

$$\mathrm{Var}[X]=E[{(X-\mu )}^2]$$

• Shorter formula:

$$\mathrm{Var}[X]=E[X^2]-E{[X]}^2$$

Calculating variance

• Discrete formula using PMF:

$$\mathrm{Var}[X]=\sum _k{(k-\mu )}^2{P}_X(k)$$

• Continuous formula using PDF:

$$\mathrm{Var}[X]={\int }_{-\infty }^{+\infty }{(x-\mu )}^2{f}_X(x)dx$$

Standard deviation

The quantity ${\sigma }_X=\sqrt{\mathrm{Var}[X]}$ is called the standard deviation of $X$.