Distributions

• For the poisson distribution, P(X = x) = (e ^ - $\lambda$ )($\lambda$ ^x)/x !. (in FB)
• The poisson distribution is used to describe the rate at which events occur.
• For a poisson distribution to be suitable, events must occur independently, singly and at a constant average rate.
• If x ~ Po($\lambda$ ) and Y ~ Po( $\mu$ ), then X + Y ~ Po( $\lambda$ + $\mu$ ).
• The mean of the poisson distribution equals the variance and is $\lambda$ .
• The poisson distribution can approximate the binomial distribution is n is large and p is small.
• When approximating the binomial distribution with the poisson, set $\lambda$ as n p.
• A geometric distribution is used to model the number of trials needed to achieve one success.
• For a geometric distribution to be a suitable model, trails must be independent of each other, and the probability of success should be the same for each trial.
• For the geometric distribution, P(X = x) = p(1 - p)^ (x - 1). (in FB)
• For the geometric distribution, P(X < x) = 1 - (1 - p)^ x and P(X > x) = (1 - p) ^(x - 1). (NOT in FB)
• The mean of the geometric distribution is E(X) = 1/p. (in FB)
• The variance of the geometric distribution is Var(X) = (1 -p)/p^2. (in FB)
• A negative binomial distribution is used to model the number of trials to achieve r number of successes.
• For a negative binomial distribution to be suitable, trials must be independent of each other and the probability of success should be the same for each trial.
• For the negative binomial distribution, P(X = x) = (x - 1) C (r - 1) p^r (1 - p)^(x - r). (in FB)
• The mean of the negative binomial distribution is E(X) = r/p. (in FB)
• The variance of the negative binomial distribution is Var(X) = r(1 - p)/p^2. (in FB)
• If X is a random sample of size n from a population with mean $\mu$ and variance $\sigma$ ^2, then the sample mean X' is approximately equal to N($\mu$ , ($\sigma$ ^2)/n).