Probability Distributions: Binomial, Poisson, and Hypergeometric

A probability distribution describes how probabilities are assigned to each possible value of a random variable .

1. Probability Distributions

A probability distribution describes how probabilities are assigned to each possible value of a random variable.

For discrete random variables, it is often given by a Probability Mass Function (PMF).
For continuous random variables, it is described by a Probability Density Function (PDF).

2. Binomial Distribution

Definition: The Binomial distribution models the number of successes in n independent trials, each with the same probability of success (p).

Random variable: $X$ = number of successes
Parameters: $n$ = number of trials, $p$ = probability of success

PMF

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0,1,2,...,n

Mean and Variance

E(X) = np, \quad Var(X) = np(1-p)

Example:

Toss a coin 5 times, count heads ( $p = 0.5$ , $n=5$ )
$P(X=3) = \binom{5}{3}(0.5)^3(0.5)^2 = 10 \cdot 0.125 \cdot 0.25 = 0.3125$

3. Poisson Distribution

Definition: The Poisson distribution models the number of events occurring in a fixed interval of time or space, given that events occur independently and at a constant rate $\lambda$ .

Random variable: $X$ = number of occurrences
Parameter: $\lambda$ = expected number of occurrences

PMF

P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k = 0,1,2,...

Mean and Variance

E(X) = \lambda, \quad Var(X) = \lambda

Example:

Cars passing through a toll booth in an hour, average 4 per hour ( $\lambda = 4$ )
Probability exactly 2 cars pass: $P(X=2) = \frac{e^{-4} 4^2}{2!} = \frac{16 e^{-4}}{2} = 8 e^{-4} \approx 0.1465$

Note: Poisson is a limit of Binomial for large $n$ and small $p$ ( $np = \lambda$ ).

4. Hypergeometric Distribution

Definition: The Hypergeometric distribution models the number of successes in a sample drawn without replacement from a finite population.

Random variable: $X$ = number of successes in the sample
Parameters:
- $N$ = population size
- $K$ = total successes in population
- $n$ = sample size

PMF

P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}, \quad \max(0,n-(N-K)) \le k \le \min(K,n)

Mean and Variance

E(X) = n \frac{K}{N}, \quad Var(X) = n \frac{K}{N} \frac{N-K}{N} \frac{N-n}{N-1}

Example:

A box contains 10 red and 15 blue balls ( $N=25, K=10$ ), draw 5 balls ( $n=5$ )
Probability exactly 2 are red: $P(X=2) = \frac{\binom{10}{2} \binom{15}{3}}{\binom{25}{5}} = \frac{45 \cdot 455}{53130} \approx 0.385$

Key: Unlike Binomial, Hypergeometric samples without replacement, so probabilities change after each draw.

5. Summary Table

Distribution	Random Variable	Parameters	PMF	Mean	Variance	Notes
Binomial	# of successes	n, p	$\binom{n}{k} p^k (1-p)^{n-k}$	$np$	$np(1-p)$	Independent trials, with replacement
Poisson	# of events in interval	$\lambda$	$\frac{e^{-\lambda} \lambda^k}{k!}$	$\lambda$	$\lambda$	Events rare, independent, constant rate
Hypergeometric	# of successes in sample	N, K, n	$\frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$	$nK/N$	$nK(N-K)(N-n)/N^2(N-1)$	Sampling without replacement

Key Points to Remember

Binomial → independent trials, fixed probability, with replacement.
Poisson → models rare events, often approximates Binomial.
Hypergeometric → no replacement, probabilities change after each draw.

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Discrete and Continuous Random Variables

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Probability Distributions: Uniform, Exponential, and Normal

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