Functions of Random variables In some case we would like to find the distribution of Y = h(X) when the distribution of X is known. Discrete case Examples 1. Let Y = aX + b , a ≠ 0 2. Let week 7

Continuous case – Examples 1. Suppose X ~ Uniform(0, 1). Let , then the cdf of Y can be found as follows The density of Y is then given by 2. Let X have the exponential distribution with parameter λ. Find the density for 3. Suppose X is a random variable with density Check if this is a valid density and find the density of . week 7

Question Can we formulate a general rule for densities so that we don’t have to look at cdf? Answer: sometimes … Suppose Y = h(X) then and but need h to be monotone on region where density for X is non-zero. week 7

Check with previous examples: 1. X ~ Uniform(0, 1) and 2. X ~ Exponential(λ). Let 3. X is a random variable with density and week 7

Theorem If X is a continuous random variable with density fX(x) and h is strictly increasing and differentiable function form R  R then Y = h(X) has density for . Proof: week 7

Summary If Y = h(X) and h is monotone then Example X has a density Let . Compute the density of Y. week 7

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 7

General Normal Distribution Let Z be a random variable with the standard normal distribution. What is the density of X = aZ + 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 7

Claim: If Y ~ N(μ,σ2) then X = aY + b has a N(aμ+b,a2σ2) 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 7

The Chi-Square distribution Find the density of X = Z2 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 7

Exercise: If find E(Y) and Var(Y). 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 V on page 576 to answer questions like: Find the value k for which . k is the 2.5 percentile of the distribution. Notation: . week 7