Distributions

Uniform Distribution: all outcomes occur with equal probability, such as a die or wheel

• PDF - 1/n where n = total # of outcomes

Binomial Distribution: experiment with n trials and two possible results where P does not change from trial to trial i.e. n coin flips

• PDF - Cⁿ_xp^x(1-p)^n-x where n = # of trials, x = number of successes, p = probablilty of success

• Expected Value - np

• Variance - np(1-p)

Poisson Distribution: estimates the # of outcomes during a set time or space

• PDF - lambda^x/x! * e^{- lambda} where lambda = mean number of occurrences, x = outcomes of interest

• Expected Value - lambda

• Variance - lambda

Hypergeometric Distribution: experiment with n trials and two possible results where P does change from trial to trial i.e. odds of 3 red balls in 5 pulls from a bag

• PDF - C^r_x * C^N-r_n-x / C^N_n where x = # of successes in trials, r = # of successes possible, N = # of elements total, n = # of trials

• Expected Value - nr/N

• Variance - (N - n)/(N - 1) * nr/N * (1 - r/N)

Negative Binomial Distributon: experiment with n trials and two possible results where P does not change from trial to trial and the trials end after a certain amount of successes i.e. odds of 5 tails before 3 heads

• PDF - C^r+x-1_r-1 p^r (1-p)^x where r = # of successes, x = # of failures

• Expected Value - r(1-p)/p

• Variance - r(1-p)/r²

Uniform Distribution (Random Variables): All points on the interval are equally likely

• PDF - 1/(B - A) for A <= x <= B, 0 elsewhere

• Expected Value - integral from -infinity to infinity xf(x)

• Variance - E(x²) - [E(x)]²

Standard Normal Distribution: standard bell-shaped curve with mean = 0 and std dev = 1

• PDF - 1/sqrt(2pi) * e^-(z)²/2 for -infinity to infinity

• CDF - P(z <= a) = Phi(a)

Exponential Distribution: time until an event occurs

• PDF - lambda * e^-lambda(x)

• CDF - 1 - e^-lambda(x)

• Expected Value - 1/lambda

• Variance - 1/lambda²

Discrete Joint Distribution