Probability Exam 2

Geometric (expected value and variance)

E[X] = 1/p Var(X) = (1 - p) / (p²)

Negative Binomial (expected value and variance)

E[X] = r/p Var(X) = r(1 - p) / (p²)

Poisson (expected value and variance)

E[X] = 𝝀 Var(X) = 𝝀

Bernoulli (expected value and variance)

E[x] = p Var(X) = p(1 - p)

Binomial (expected value and variance)

E[x]=np Var(x) = np(1 - p)

Poisson(𝝀)

𝝀 = np x = 0, 1, 2, ...

Binomial Random Variable

X denotes the *numbers of successes* in *n* independent trials where each trial has a probability of success of *p*

Bernoulli random variable

X denotes whether a trial that results in a success with probability *p* is a success or not

Geometric Random Variable

X is *numbers of trials needed* to obtain a success when each trial is independent with a probability of success of *p*

Hypergeometric Random Variable

X is defined to be the numbers of successes obtained in a *random sample, n* selected *without replacement* from a finite *population of N* elements that contains *m* successes and N-m failures

Negative Binomial Random Variable

X is numbers of trials needed to obtain to of *r* success when each trial is independent with a probability of success of p

Poisson Random Variable

X is used to denote the numbers of events that occur when these events are independent (or weakly dependent) and each have a small probability of occurrence

Binomial(n,p)

n = numbers of trials p = probability of success k = numbers of successful trails

Geometric(p)

p = probability of success k = first successful trial

Negative Binomial(r, p)

r = numbers of successes (failures) k = r, r + 1, r + 2, ...