Random Variables

The quantities associated with random experiments

Definition

• Real valued function where the domain is the sample space
• $X:S\to \mathbb{R}$
• The range of $X$ is denoted $\mathcal{X}$
• Discrete if range is discrete, continuous if continuous

Expectation

• Expectation
  • The expected value of the random variable
  • Notated $\mathbb{E}(X)$
  • Commonly referred to as the mean (denoted $\mu$)
  • Properties
    • $\mathbb{E}(aX+b)=a\mathbb{E}(X)+b$ (linear property)
• Variance
  • ${\sigma }^2=V(X)=\mathbb{E}[(X-\mu {)}^2]$
  • Average squared distance between values of $X$ and $\mathbb{E}(X)$
  • Problem: Unit is not interpretable
  • Properties
    • $V(aX+b)=a^{2}V(X)$
    • $V(X)=\mathbb{E}(X^2)-(\mathbb{E}(X){)}^2$
    • If $Y_1,\dots ,Y_n$ independent, $V(\sum _{i=1}^n{Y}_i)=\sum _{i=1}^{n}V(Y_i)$
• Standard Deviation
  • $\sigma =SD(X)=\sqrt{V(X)}$
  • Variance but maintains original unit of measure
• Chebyshev Inequality
  • For any positive $k$, $P(|X-\mu |<k\sigma )\ge 1-\frac{1}{k^2}$
  • Tells us how grouped up something is to its mean
  • This is just a lower bound, often an underestimate

Types

Discrete Random Variables Continuous Random Variables

Moment Generating Function

• kth moment = $\mathbb{E}[X^k]$
• moment generating function: $M_X(t)=\mathbb{E}[e^{tX}]$ for every $t\in \mathbb{R}$
• kth moment can be obtained by taking the kth derivative with respect to $t$ at $t=0$
  • $\mathbb{E}[X^k]=\frac{d^k}{d{t}^k}{M}_X(t){|}_{t=0}$
• MGF is unique for a distribution
• Can derive MGF from a linear transform
  • $Y=aX+b$
  • $M_Y(t)=e^{bt}{M}_X(at)$

Joint Mass Functions

• Discrete
  • $p_X(x)(x)=P(X=x)={\sum }_{y\in \mathcal{Y}}P(X=x,Y=y)$
• Conditional: $p_{Y|X=x}(y)=P(Y=y|X=x)=\frac{P(X=x,Y=y)}{P(X=x)}$