Binomial Probability Distribution

Study Notes

The expected value of a binomial distribution, E(X) = np
The variance of a binomial distribution, V(X) = np(1-p)
The standard deviation of a binomial distribution, σ(X) = √np(1-p)
Binomial distribution is used when counting the number of successes in n trials
Example: Tossing a fair coin 10 times, what is the probability of obtaining 3 heads?

The variance of a Poisson distribution, V(X) = E(X) = λ
The expected value of X² in a Poisson distribution, E(X²) = E(X) + (E(X))²
Poisson distribution is used when counting the average number of occurrences per unit of time/space
Example: A phone operator receives an average of 2 calls per minute. Assume the number of calls follows a Poisson distribution. What is the probability of receiving 10 calls in 5 minutes?

The probability of success in a geometric distribution, p(k) = (1-p)^(k-1) * p
The variance of a geometric distribution, V(X) = (1-p)/p²
The expected value of a geometric distribution, E(X) = 1/p
Geometric distribution is used when counting the number of trials until the first success
Example: A certain exam is either difficult (p=90/92) or fair (p=2/92). Exams are taken one after the other. What is the probability that at least 4 exams will occur before the first fair one?

The expected value of a negative binomial distribution, E(X) = r/p
The variance of a negative binomial distribution, V(X) = r(1-p)/p²
The standard deviation of a negative binomial distribution, σ(X) = [√(r(1-p))] /p
The expected value of Y, where Y is the number of failures, E(Y) = r(1-p)/p
The probability of success in a negative binomial distribution, p(x) = x-1Cr-1 * p^r * (1-p)^(x-r)
Alternative definition: Let Y be the number of failures, then p(y) = y+r-1Cr-1 * p^r * (1-p)^y
Example: In tossing a fair dice repeatedly, find the probability of getting the third "1" on the 7th toss