Discrete Random Variables 2

Jan 17, 2024

---

The Big Three

The following is a list of essentail formulaas for the three most important discrete distributions: \(\textbf{binomial, geometric, and Poisson.}\)

• \(\textbf{Binomial distribution } b(n, p):\)
  • \(n \text{ (positive integer), } p \: (0 \leq p \leq 1)\)
  • \(\textbf{p.m.f: } f(x) = {n \choose x}p^x (1 - p)^{n - x} \:\: (x = 0, 1, 2, ..., n)\)
  • \(\textbf{Expectation and variance: } \mu = np, \:\: \sigma^2 = np(1 - p)\)
  • \(\textbf{Arises as: }\) Distribution of number of successes in success/failure trials (Bernoulli trials)
• \(\textbf{Geometric distribution:}\)
  • \(p \: (0 < p < 1)\)
  • \(\textbf{p.m.f: } f(x) = (1 - p)^{x - 1} p \:\: (x =1, 2, ...)\)
  • \(\textbf{Expectation and variance: } \mu = 1/p, \: \sigma^2 = (1 - p)/p^2\)
  • \(\textbf{Geometric series: } \overset{\infty}{\underset{n=0}{\sum}} r^n = \frac{1}{1 - r} (| r | < 1)\)
  • \(\textbf{Arises as: }\) Distribution of trial at which the first success occurs in success/failure trial sequence
• \(\textbf{Poisson disctribution:}\)
  • \(\lambda > 0\)
  • \(\textbf{p.m.f: } f(x) = \mathcal{e}^{-\lambda \frac{\lambda^x}{x!}} \:\: (x = 0, 1, 2, ...)\)
  • \(\textbf{Expectation and variance: }\overset{\infty}{\underset{n=0}{\sum}} \frac{\lambda^n}{n!} = \mathcal{e}^\lambda\)
  • \(\textbf{Arises as: }\) Distribution of number of occurrences of rare events, such as accidents, insurance claims, etc.

Other discrete distributions

1. \(\textbf{Hypergeometric distribution:} f(x) = \frac{ {N_1 \choose x} {N_2 \choose n - x} }{N \choose n}\), \(x = 0, 1, ..., N_1, n - x \leq N_2\)

2. \(\textbf{Negative binomial distribution:} f(x) = {x - 1 \choose r - 1}(1 - p)^{x - r}p^r\), \(\quad x = r, r + 1, ...\)

Binomial coefficients

• \(\textbf{Definition: } \text{For } n = 1, 2, ... \text{ and } k = 0, 1, ..., n, {n \choose k} = \frac{n!}{k!(n - k)!}\)

• \(\textbf{Alternate notations: } {}_n C_k \text{ or } C(n, k)\)

• \(\text{Other definition:} {n \choose k} = \frac{n(n-1)...(n - k + 1)}{k!}\)

• \(\textbf{Symmetry property: } {n \choose k} = {n \choose n - k}\)

• \(\textbf{Special cases: } {n \choose 0} = {n \choose n} = 1, \:\: {n \choose 1} = {n \choose n - 1} = n\)

• \(\textbf{Binomial Theorem: } (x + y)^n = \overset{n}{\underset{k=0}{\sum}} {n \choose k} x^k y^{n - k}\)

• \(\textbf{Binomial Theorem, special case: } (x + y)^n = \overset{n}{\underset{k=0}{\sum}} {n \choose k} p^k(1 - p)^{n - k} = 1\)

• \(\textbf{Combinatorial Interpretations: } {n \choose k} \text{ represents }\)

1. the number of ways to select \(k\) objects out of \(n\) given objects (in the sense of unordered samples wihtout replacement)

2. the number of \(k\)-element subsets of an \(n\)-element set

3. the number of \(n\)-letter \(\text{HT}\) sequences with exactly \(k\) \(\text{H's}\) and \(n - k\) \(\text{T's}\)

• \(\textbf{Binomial distribution: }\) Given a positive integer \(n\) and a number \(p\) with \(0 < p < 1\), the binomial distribution \(b(n, p)\) is the distribution with density (p.m.f) \(f(x) = {n \choose x} p^x(1 - p)^{n - x}\), for \(x = 0, 1, ..., n\).