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The density function of a normal distribution with mean µ and standard deviation σ : ! ! = ! !!! ! exp − !!! ! !! ! , where -∞ < x < ∞, ∞< µ < ∞, 0 < σ <∞. For the standard normal distribution, µ = 0 and σ = 1. Another distribution that a bona fide statistician cannot live without is the gamma distribution. Its density function is: ! ! = ! !(!)! ! ! !!! ! !!/! where x > 0; α , β > 0. The gamma distribution includes as special cases the exponential distribution (α = 1) and the chi-square(d) distribution (α = n/2, β = 2); a limiting case includes the normal distribution (α → ∞). The uniform distribution on the unit interval has density function g(x) = 1 0 < x < 1. If the random variable X has this uniform distribution, then Y = F -1 (X) has distribution function F. A useful version of the Central Limit Theorem states that given a random sample X1, X2, …, Xn from a distribution with mean µ and standard deviation σ, the distribution of the sample mean approaches the normal distribution with mean µ and standard deviation σ / √n . (The random sample wording implies that the Xi’s are independent and identically distributed; the variance σ 2 is assumed to be finite). Mean: E(X) = µ Variance: Var (X) = E[(X-µ)2] = σ 2 (standard deviation σ has same units as X). Skewness: E[(X-µ)3/σ 3] (zero for symmetric distributions) Kurtosis: E[(X-µ)4/σ 4] (equals 3 for the normal distribution) The probability mass function of the discrete binomial distribution is given by: ! ! = ! = Note that ! ! = ! ! !! !! !!! ! ! ! (1 − !)!!! for k = 0, 1, …, n; 0 < p < 1. , where n! = n × (n-1) × (n-2) × … × 2 × 1.