Title 31 Common Discrete Probability Distributions.pdf ✅

COMMON DISCRETE PROBABILITY DISTRIBUTIONS 1. BINOMIAL DISTRIBUTION The probability distribution describes the probability of x successes in n Bernoulli trials. • Bernoulli Process – repeated independent trials with only two outcomes: success and failure. • Probability problems involving the term "with replacement" use the Bernoulli Process. [DERIVATION] If there are x successes in n trials, then there are n − x failures. The number of ways to distribute such is ( n x ) = n! x! (n − x)! Since the outcomes are independent, then multiply the probabilities: p x ⋅ q n−x Where p + q = 1, p is the probability of success, and q is that of failure. The binomial distribution, therefore, is b(x; n, p) = ( n x ) p xq n−x • This formula is also similar to the binomial theorem based on the expansion of (p + q) n . The mean of the binomial distribution is μ = np While the standard deviation is given by σ = √npq Advanced discussions of algebra, calculus, and statistics are the basis of the derivation of the two formulas. 1.1. Multinomial Distribution It is an extension of the binomial theorem where the events are still under the Bernoulli Process, but the outcomes are more than 2. The multinomial probability distribution is given by f(x1, x2, ⋯ xk; p1, p2, ⋯ pk, n) = n! x1! x2! x3! ⋯ xk! ⋅ p1 x1p2 x2 ⋯ pk xk The multinomial distribution is also similar to the multinomial theorem.

3. Negative Binomial Distribution The Negative Binomial Distribution is also a special case of the binomial distribution. Still, it differs according to the following condition: The kth success in the xth trial. [DERIVATION] Since the xth trial is limited to a “success,” the first k − 1 successes will be distributed on the first x − 1 trials. From binomial distribution, ( x − 1 k − 1 ) ⋅ p k−1 ⋅ q x−k ⋅ p Therefore, the negative binomial distribution is ( x − 1 k − 1 ) ⋅ p k ⋅ q x−k 4. Geometric Distribution It is a special case of the negative binomial distribution where it satisfies the condition: The first success in x trials. [DERIVATION] Since the first success is on the xth trial, then the first x − 1 trials are failures. Therefore, the geometric probability is pq x−1 The mean of a geometric distribution is μ = 1 p While its variance is σ 2 = 1 − p p 2