Functions of Random variables

1. 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

2. 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

3. 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

4. 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

5. 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

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

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

8. 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

9. 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

10. 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

11. 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