Probability and Statistics: Discrete Random Variables

Probability Mass Function (pmf)

A function that provides the probabilities of a discrete random variable, assigning a probability to each possible value the variable can take.

P(X = x)

notation used when solving discrete distributions

Cumulative Distribution Function (cdf)

Gives the probability that a discrete random variable is less than or equal to a specific value.

P(X ≤ x)

Cumulative Distribution Function setup

h(x) * p(x)

E[h(X)] =

a * E(X) + b

E(aX + b)

E(X²) - [E(X)]²

V(X) =

a² * V(X)

V(aX + b) =

Square root of Variance

How to get Standard Deviation

Bernoulli Distribution

Discrete Distribution used to model situations with a single trial and two possible outcomes

p

P(Success) =

1 - p

P(Failure) =

P(X = x) = p^x * (1 - p)^1-x

Probability mass function of Bernoulli

p

Mean of Bernoulli

p(1-p)

Variance of Bernoulli

Binomial Distribution

A probability distribution that summarizes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is described by two parameters: the number of trials and the probability of success.

P(X = x) = (n x) * p^x * (1-p)^n-x

The probability mass function of a Binomial Distribution, where n is the number of trials, x is the number of successes, and p is the probability of success.

np

The expected value of a Binomial Distribution, representing the average number of successes in n independent Bernoulli trials each with success probability p.

np(1-p)

The variance of a Binomial Distribution, indicating the measure of variability around the expected value.

Negative Binomial Distribution

Discrete Distribution used to answer the question how many trials it takes until a given number of successes is reached

P(X = x) = (x-1 r-1) p^r (1-p)^x-r

Probability mass function of negative binomial random variable

r/p

Mean for Negative Binomial Random Distribution

r(1-p)/p²

Variance for Negative Binomial Distribution

Geometric Distribution

Discrete Distribution used when asking for the number of trials before the first success

P(X = x) = (1-p)^x-1 * p

Probability Mass Function for Geometric Random Variable

1/p

is the expected number of trials in a Geometric Distribution before the first success.

1-p/p²

Variance of geometric distribution

x * p(x)

E(X) =