Functions and Transformations of Random Variables

1. Section 09 Functions and Transformations of Random Variables

2. Transformation of continuous X • Suppose X is a continuous random variable with pdf and cdf • Suppose is a one-to-one function with inverse ; so that • The random variable is a transformation of X with pdf: • If is a strictly increasing function, then • and then

3. Transformation of discrete X • Again, • Since X is discrete, Y is also discrete with pdf • This is the sum of all the probabilities where u(x) is equal to a specified value of y

4. Transformation of jointly distributed X and Y • X and Y are jointly distributed with pdf • and are functions of x and y • This makes and also random variables with a joint distribution • In order to find the joint pdf of U and V, call it g(u,v), we expand the one variable case • Find inverse functions and so that and • Then the joint pdf is:

5. Sum of random variables • If then • If Xs are continuous with joint pdf • If Xs are discrete with joint pdf

6. Convolution method for sums • If X1 and X2 are independent, we use the convolution method for both discrete & cont. • Discrete: • Continuous:

7. Sums of random variables • If X1, X2, …, Xn are random variables and • If Xs are mutually independent

8. Central Limit Theorem • X1, X2, …, Xn are independent random variables with the same distribution of mean μ and standard deviation σ • As n increases, Yn approaches the normal distribution • Questions asking about probabilities for large sums of independent random variables are often asking to use the normal approximation (integer correction sometimes necessary).

9. Sums of certain distribution • This table is on page 280 of the Actex manual • There are more than these but these are the most common/easy to remember

10. Distribution of max or min of random variables • X1 and X2 are independent random variables

11. Mixtures of Distributions • X1 and X2 are independent random variables • We can define a brand new random variable X as a mixture of these variables! X has the pdf • Expectations, probabilities, and moments follow this “weighted-average” form • Be careful! Variances do not follow weighted-average! Instead, find first and second moments of X and subtract

12. Sample Exam #101 The profit for a new product is given by Z = 3X – Y – 5 . X and Y are independent random variables with Var(X)=1 and Var(Y)=2. What is the variance of Z? A) 1 B) 5 C) 7 D) 11 E) 16

13. Sample Exam #102 A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 10. The company will begin using the second generator immediately after the first one fails. What is the variance of the total time that the generators produce electricity?

14. Sample Exam #103 In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be independent, exponentially distributed random variables with respective means 1.0, 1.5, and 2.4 Determine the probability that the maximum of these losses exceeds 3.

15. Sample Exam #142 An auto insurance company is implementing a new bonus system. In each month, if a policyholder does not have an accident, he or she will receive a $5 cash-back bonus from the insurer. Among the 1000 policyholders of the auto insurance company, 400 are classified as low-risk drivers and 600 are classified as high-risk drivers. In each month, the probability of zero accidents for high-risk drivers is .8 and the probability of zero accidents for low-risk drivers is .9 Calculate the expected bonus payment from the insurer to the 1000 policyholders in one year.

16. Sample Exam #123 You are given the following information about N, the annual number of claims for a randomly selected insured: Let S denote the total annual claim for an insured. When N=1, S is exponentially distributed with mean 5. When N>1, S is exponentially distributed with mean 8. Determine P(4<S<8).