Random Variables
With $X$ being a discrete random variable, $f(x)=P(X=x)$ for each $x$ within the domain of $X$ is called the probability distribution of $X$

• or the probability mass function (PMF)

A function $f(x)$ can serve as a PMF of a discrete random variable $X$ if and only if:

1. $f(x)\ge 0,\mathrm{\forall }x\in X$
2. $\sum _{x}f(x)=1$

PMFs can be represented by simple histograms or bar charts

Cumulative Distribution Function of $X$ (CDF):

where $f(t)$ is the probability distribution of the discrete random variable $X$ at $t$

The values $F(x)$ of $X$ need to satisfy:

1. $F(-\mathrm{\infty })=0,\text{ and }F(\mathrm{\infty })=1$
2. $a<b\to F(a)\le F(b)$ for any $a$ and $b$ (non-decreasing at every point)

If the range of a random variable $X$ consists of $k$ values $x_1<x_2<\cdots <x_k$ , then

• $f(x_1)=F(x_1)$
• $f(x_i)=F(x_i)-F(x_{i-1}),\mathrm{\forall }i=1,2,\dots k$

Probability Densities

Probability distribution of a random variable

• probability mass function
  • P(X = x)
  • cumulative distribution function
    • $F(x)=P(X\ge )$
• a listing of all possible values of X and their probabilities of occurring
• $P(X=x)=\left(\begin{array}{c}2\\ x\end{array}\right){0.03}^x{0.97}^{2-x}$
  • for x = 0,1,2
• 
• expectation and variance
  • expected value/expectation
    • the theoretical mean of the random variable, or equivalently, the mean of its probability distribution
      • real distribution is never gonna be exactly = the theoretical one
      • 
    • E(x)
      • values x probability of occurring
      • $$E(X)=\sum _{allx}xp(x)$$
      • Mu = theoretical mean = E(x)
    • expectation of a function g(X)
      • $$E[g(X)]=\sum _{allx}g(x)p(x)$$
    • variance of a disc random variable
      • the expectation of its squared distance from the mean
      • $$Var(X)=E[(X-\mathrm{\mu }{)}^2]=\sum _{allx}(x-\mathrm{\mu }{)}^{2}p(x)$$
      • handy relationship
        • theoretically the same, often easier to calculate
        • $$E[(X-\mathrm{\mu }{)}^2]=E(X^2)-[E(X){]}^2$$
          • $E(x)=\mu$
          • $\text{E[(x-mu)^2] = E[x^2-2xmu+mu { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#2">\#2</a>} ]=E(x^2)-2E(xmu)+E(mu^2)}$
          • $=E(x^2)-2E(x)\cdot \mu +{\mu }^2$
          • $=E(x^2)-2\mu \mu +{\mu }^2=E(x^2)-{\mu }^2=E(x^2)-[E(x){]}^2$
    • sd of a random variable(σ)
      • sqrt of variance (Var(X))
        Hypergeometric Distribution
        Bernoulli Distribution