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Discrete Probability Distributions

This comprehensive guide delves into discrete probability distributions, focusing on random variables and expected values, including the mean (E(X)). It presents essential formulas, such as the calculation of variance (Var(Y)) and standard deviation (s), along with practical examples. The binomial distribution with its application in Bernoulli trials is discussed, illustrating how to calculate the probability of successes in trials. Cumulative probabilities and useful resources like binomial calculators are also highlighted for further study, helping students grasp these fundamental statistical concepts.

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Discrete Probability Distributions

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  1. Discrete Probability Distributions

  2. Random Variables

  3. Expected Value (mean, average) Μ = E(X) = Σ value(x) x probability(x)

  4. Expected value = 0.5 x (-1) + 0.5 x (+1) = 0.

  5. Example: Lottery of 1,000 tickets, with the following payout structure, has an E(x) = $1.00.

  6. Variance s2 (or “spread”) and Standard Deviation s

  7. Calculating the Variance s2 and the Standard Deviation s. Var(Y) = s2 = Σ[(y – E(Y))2 x P(y)] Or Var(Y) = s2 = Σ[P(y) x Y] – E(Y)2 Sdev s = (s2)0.5

  8. Mathematics Factorials ! n! = n x (n – 1) x (n – 2) …. x 1 Example: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720.

  9. Binomial Distribution • n Bernoulli trials, where each trial only has two possible outcomes, like a coin toss (heads or tails) or baby (boy or girl). • p = the probability of success in each trial, and (1 – p) is the probability of failure. • The probability of x success in n trials is ……

  10. Example: a die is rolled exactly n = 5 times. What is the probability of rolling exactly x = 2 sixes? (Note the probability of rolling a 6 is P(six) = 1/6 = 0.166667.)

  11. Binomial Calculators (online) http://stattrek.com/tables/binomial.aspx Or, using MS Excel, go to Formulas/More Functions/Statistical/BINOMDIST

  12. Cumulative Binomial Probabilities • As before, a question can be about the probability of exactly x successes in n trials. P(X = x). • But the question can also be about the (Cumulative) probability of getting x or less successes. P(X ≤ x). • Example: P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). • The probability of getting at least 4 successes in n trials = 1 – P(X ≤ 3).

  13. Binomial Distribution Statistics Mean = np Variance = np(1 – p)

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