13 Lecture 12, Feburary 02, 2024
13.1 Hypergeometric distribution
Definition 13.1 (Hypergeometric distribution) Consider a population that consists of \(N\) objects, of which \(r\) are considered successes and the remaining \(N-r\) are considered failures. Suppose that a subset of size \(n\) (with \(n\leq N\)) is drawn from the population WITHOUT REPLACEMENT. Let \(X\)=Number of successes obtained, then we say \(X\) follows a hypergeometric distribution with parameters \((N,r,n)\). We sometimes write \(X \sim hyp(N,r,n)\) or \(X\sim HG(N,r,n)\).
13.1.1 Range of the Hypergeomnetric distribution
- We cannot have more successes than there are (\(r\)) \(\Rightarrow\) \(x\leq r\)
- We cannot have more successes than trials (\(n\)) \(\Rightarrow\) \(x\leq n\)
- When there are more trials than failures (\(n>(N-r)\)) we will for sure have at least \(n-(N-r)\) successes \(\Rightarrow\) \(x\geq \max\{0, n-(N-r)\}\).
- Overall, we have \(\max\{0, n-(N-r)\} \leq x \leq \min\{r,n\}\).
13.1.2 Probability function of hypergeometric distribution
- Total number of of subsets of size \(n\): \(\binom{N}{n}\).
- Number of ways to select \(x\) successes out of \(r\) successes: \(\binom{r}{x}\).
- Number of ways to choose remaining \(n-x\) failures from \(N-r\) failures: \(\binom{N-r}{n-x}\).
- Thus, \[f(x) = \frac{\binom{r}{x}\binom{N-r}{n-x}}{\binom{N}{n}},\] where \(\max\{0, n-(N-r)\} \leq x \leq \min\{r,n\}\) and 0 otherwise.
13.2 Bernoulli and Binimial distributions
Definition 13.2 (Bernoulli trail) A Bernoulli trial with probability of success \(p\) is an experiment that results in either a success or failure, and the probability of success is \(p\).
Definition 13.3 (Bernoulli distribution) If a random variable \(X\) represents the number of successes in a Bernoulli trial with probability of success \(p\), it follows the Bernoulli distribution, and we denote it as \[ X \sim Bernoulli(p). \]
13.2.2 Bernoulli
If \(X\sim Bern(p)\), the pf of \(X\) is \[ f_X(0)=1-p,\quad f_X(1)=p,\quad f_X(x)=0\,\,\text{ for }x\not\in\{0,1\}\] and the cdf is \[ F_X(x)=\begin{cases} 0, &\text{ if }x<0, \\ 1-p, &\text{ if } 0\leq x < 1,\\ 1, &\text{ if }x\geq 1\end{cases}\]
13.2.3 Binomial
Definition 13.4 (Bernoulli distribution) If we have \(N\) independent runs and record the numbers of successes obtained in these \(n\) runs, then \(X\) is said to have a binomial distribution, denoted by \(X\sim Bin(n,p)\).
Note: The probability function of \(X\sim Bin(n, p)\) is \[ f(x) = \binom{n}{x} p^x(1-p)^{n-x},\quad x=0,1,2,\dots,n\] and 0 otherwise.
13.3 Relationship and difference between binomial and hypergeometric
- Binomial and hypergeometric distributions are fundamentally different!
- In Binomial models, we pick WITH replacement, in the hypergeometric model WITHOUT replacement.
- If \(N\) is large and \(n\) is small, the chance we pick the same object twice is small.
- Thus, letting \(r/N=p\), \(X\sim Hyp(N,r,n)\) and \(Y\sim Bin(n,p)\), then we can APPROXIMATE \[ P(X \leq k) \approx P(Y\leq k).\] (more precisely, in the limit as \(N\rightarrow\infty\) with \(r/N\rightarrow p\) (the ratio of successes converges to the success probability).
- See pages 86/87 and later in the course for more.
- We’ll see an example next time.