Random Variables

Random Variable: Function $X:\Omega \to \mathcal{R}$ ($\Omega$ is a sample space, $\mathcal{R}$ is a measurable space)

• Discrete Random Variable: $\text{range}(X)$ is countable or countably infinite
• Continuous Random Variable: $\text{range}(X)$ is uncountably infinite

$P(X=x) = p(x)$: The probability that $X$ takes the value $x$.

• Probability Distribution: Formula, table, or graph that provides $p(x)$ for all $x \in \Omega$

Discrete Probability Distribution

• $\forall x \in \Omega, 0 \le p(x) \le 1$
• $\sum_xp(x)=1$

Expected Value / Mean: $E(X)= \mu = \sum_x x\cdot P(X=x)$

• $E(c) =c$ where $c$ is a constant
• $E[c\cdot g(X) + k]=c \cdot E[g(X)] + k$
• $E[g_1(X)+\dots+g_n(X)]=E[g_1(X)]+ \dots + E[g_n(X)]$

Variance: $V(X) =$ Expected value $E[(X-\mu)^2]$

• $V(X)=E(X^2)-E(X)^2$

Standard Deviation: $S(X)= |\sqrt{V(x)}|$

Distribution Function: $F: \mathcal{R} \to [0, 1]$

• $F(a)=P(X\le a) \text{ for } a \in (-\infin, \infin)$

Probability Distributions

Bernoulli Distribution