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. 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