Course Notes - Department of Mathematics and Statistics

BSummary of Formulae1. Normal DistributionIf X is a normal random variable with parameters µ X (mean) and σ 2 X(variance)• Mean: µ X• Standard deviation: σ X =√σ 2 XA standard normal random variable Z has mean µ Z = 0 and σ 2 Z = 1.To transform a normal random variable X into a standard normal(and vice versa):2. Binomial DistributionZ = X − µ Xσ Xand X = Zσ X + µ X .If X is a binomial random variable with n trials and probability πthen• Mean: µ X = nπ• Standard deviation: σ X = √ nπ(1 − π)• If nπ and n(1 − π) are both greater than 5, then X is approximatelynormally distributed with mean µ X and variance σ 2 X .3. Distributions of Statistics• The mean ¯X of a random sample of size n has mean µ ¯X = µ Xand standard deviation σ ¯X = √ σ Xn.• The sample proportion P computed from a binomial distributionwith parameters √n and π has a mean of µ P = π and standardπ(1−π)deviation σ P =n. If nπ and n(1 − π) are both greaterthan 5, then P will be approximately normally distributed.• The distribution of the difference between two sample means ¯X 1 −¯X 2 has a mean of µ ¯X1 − ¯X 2= µ 1 − µ 2 and a standard deviation of√σ ¯X1 − ¯Xσ12=2 n 1+ σ2 2n 2.271