More about Normal Distributions The Standard Normal Gaussian

More about Normal Distributions • The Standard Normal (Gaussian) random variable, X ~ N(0, 1), has a density function given by • Exercise: Prove that this is a valid density function. • The cdf of X is denoted by Φ(x) and is given by • There are tables that provide Φ(x) for each x. However, Table 4 in Appendix 3 of your textbook provides 1 - Φ(x). • What are the mean and variance of X? E(X) = Var(X) = week 6 1

General Normal Distribution • Let Z be a random variable with the standard normal distribution. What is the density of X = a. Z + b , for ? • Can apply change-of-variable theorem since h(z) = az + b is monotone and h-1 is differentiable (assuming a ≠ 0). The density of X is then • This is the non-standard Normal density. • What are the mean and variance of X? • If Y ~ N(μ, σ2) then . week 6 2

• Claim: If Y ~ N(μ, σ2) then X = a. Y + b has a N(aμ+b, a 2σ2) distribution. Proof: • The above claim shows that any linear transformation of a Normal random variable has another Normal distribution. • If X ~ N(μ, σ2) find the following: week 6 3

The Chi-Square distribution • Find the density of X = Z 2 where Z ~ N(0, 1). • This is the Chi-Square density with parameter 1. Notation: . • χ2 densities are subsets of the gamma family of distributions. The parameter of the Chi-Square distribution is called degrees of freedom. • Recall: The Gamma density has 2 parameters (λ , α) and is given by α – the shape parameter and λ – the scale parameter. week 6 4

• The Chi-Square density with 1 degree of freedom is the Gamma(½ , ½) density. • Note: • In general, the Chi-Square density with v degrees of freedom is the Gamma density with λ = ½ and α = v/2. • Exercise: If find E(Y) and Var(Y). • We can use Table 6 in Appendix 3 to answer questions like: Find the value k for which. k is the 2. 5 percentile of the distribution. Notation: . week 6 5