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: $binomial, geometric, and Poisson.$

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

Other discrete distributions

$Hypergeometric distribution: f (x) = \frac{(\binom{N_{1}}{x}) (\binom{N_{2}}{n - x})}{(\binom{N}{n})}$ , $x = 0, 1, . . ., N_{1}, n - x \leq N_{2}$
$Negative binomial distribution: f (x) = (\binom{x - 1}{r - 1}) (1 - p)^{x - r} p^{r}$ , $x = r, r + 1, . . .$

Binomial coefficients

$Definition: For n = 1, 2, . . . and k = 0, 1, . . ., n, (\binom{n}{k}) = \frac{n!}{k! (n - k)!}$
$Alternate notations:_{n} C_{k} or C (n, k)$
$Other definition: (\binom{n}{k}) = \frac{n (n - 1) . . . (n - k + 1)}{k!}$
$Symmetry property: (\binom{n}{k}) = (\binom{n}{n - k})$
$Special cases: (\binom{n}{0}) = (\binom{n}{n}) = 1, (\binom{n}{1}) = (\binom{n}{n - 1}) = n$
$Binomial Theorem: (x + y)^{n} = \overset{n}{\sum_{k = 0}} (\binom{n}{k}) x^{k} y^{n - k}$
$Binomial Theorem, special case: (x + y)^{n} = \overset{n}{\sum_{k = 0}} (\binom{n}{k}) p^{k} (1 - p)^{n - k} = 1$
$Combinatorial Interpretations: (\binom{n}{k}) represents$

the number of ways to select $k$ objects out of $n$ given objects (in the sense of unordered samples wihtout replacement)
the number of $k$ -element subsets of an $n$ -element set
the number of $n$ -letter $HT$ sequences with exactly $k$ $H’s$ and $n - k$ $T’s$

$Binomial distribution:$ 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) $f (x) = (\binom{n}{x}) p^{x} (1 - p)^{n - x}$ , for $x = 0, 1, . . ., n$ .