Discrete Distribution PMF’s

Question 1

Discrete Uniform

Answer 1

px(x) = 1/(b-a+1)

With “b” and “a” representing the upper and lower bounds of the distribution

Probability of each outcome (all equal)

Question 2

Binomial

Answer 2

px(x) = nCx p^x (1-p)^n-x

With “n” representing the number of trials or a sample size with two outcomes, “p” representing the probability of success, and “x” used to find the number of successes/goal

Gives the probability of “x” successes of one outcome (heads vs tails) with “n” trials

Question 3

Hypergeometric

Answer 3

px(x) = [(mCx * n-mCn-x)/(NCn)]

With “n” the number of dependent trials, “N” the population size, “m” the number of successes, and “x” the goal of the number of successes

Sampling without replacement

Gives the probability of “x” successes with “n” draws from “N” population size without replacement with “m” total successes possible

Question 4

Geometric

Answer 4

px(x) = (1-p)^(x-1) * p

Number of trials to get to 1st success

py(y) = (1-p)^y * p

Number of failures before 1st success

With “p” equal to the probability of success, “x” trials to get to 1st success, and “y” number of failures trials before 1st success

Gives probability of 1st success happening on the “x” trial

Question 5

Negative Binomial

Answer 5

px(x) = (x-1)C(r-1) p^r (1-p)^(x-r)

With “r” being the number of desired successes, “p” the probability of success, and “x” is the number of trials

Gives probability of “r” successes happening in “x” trials

Question 6

Poisson

Answer 6

px(x) = (e^(-λ) * λ^x) / (x!)

With “λ” representing the mean of x, and “x” representing the number of occurrences of an event

Gives the probability of “x” occurrences of an event during a fixed interval with mean “λ”