Probability Distribution

In Probability Theory, a Probability Distribution is a Function that maps a Sample Space to a Probability Space.

Sampling with Replacement

aka Binomial Distribution

• Of a sample drawn from a finite population, drawing $n$ objects with replacement from the set of typed objects, containing a objects of type 1 and b objects of type 2, and get exactly $k$ of type 1

$$\begin{array}{rl}P(\text{k type 1 were picked})=& \\ (\text{\# ways to choose k 1s and n-k 2s})\cdot P(\text{choose k 1s in a row, then n-k 2s in a row})& =\\ C(n,k){\left(\frac{a}{a+b}\right)}^k{\left(\frac{b}{a+b}\right)}^k& \end{array}$$

Sampling without Replacement

aka Hypergeometric Distribution

• Of a sample drawn from a finite population, drawing $n$ objects without replacement from the set of typed objects, containing a objects of type 1 and b objects of type 2, and get exactly $k$ of type 1

$$\begin{array}{rl}P(\text{k type 1 were picked})=& \\ \frac{(\text{\# ways to choose k 1s})(\text{choose n - k 2s})}{\text{\# ways to choose n out of a + b}}=& \\ \frac{C(a,k)C(b,n-k)}{C(a+b,n)}& \end{array}$$