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1 Contents 1 Contents 6 Distributions of Functions of Random Variables Transformation of Discrete r.v.s Method of Distribution Functions Method of Transformations Method of Moment Generating Functions

2 6 Distributions of Functions of Random Variables 2 6 Distributions of Functions of Random Variables Let Y be a r.v. and U = h(y ) be a r.v. defined as a function of Y. Objective: given the distribution of Y, derive the distribution of U. Four approaches: 1. Direct calculation of the pmf (for discrete r.v.s). 2. Method of distribution functions (for continuous r.v.s): Integrate over the region in the y space that corresponds to the event {U u} to obtain the CDF of U: F U (u) = P (U u). Differentiate F U (u) with respect to u to obtain the pdf of U: f U (u). 3. Method of transformations (for continuous r.v.s): Derive the distribution of U by methods akin to the method of

3 6 Distributions of Functions of Random Variables 3 substitution (or change of variable) in calculus 4. Method of moment-generating functions: Make use of the uniqueness property of mgfs and other properties to deduce the distribution of functions of r.vs. 6.1 Transformation of Discrete r.v.s To determine the pmf of U = h(y ), where Y is a discrete r.v., 1. determine the support of U: unique values of h(y) for all possible values of Y ; 2. determine P (U = u) for each possible value of U.

4 6 Distributions of Functions of Random Variables 4 Example Suppose Y follows a geometric distribution with pmf P (Y = y) = (1 p) y 1 p, y = 1, 2, Determine the pmf of U = Y 1: the number of failures before the first success is observed.

5 6 Distributions of Functions of Random Variables 5 Example (Sum of two independent discrete random variables) Suppose X Binomial(2, 0.5) and Y Binominal(1, 0.2) are independent. Determine the pmf of U = X + Y.

6 6 Distributions of Functions of Random Variables Method of Distribution Functions Procedure: 1. Integrate over the region in the y space that corresponds to the event {U u} to obtain the CDF of U: F U (u) = P (U u) 2. Differentiate F U (u) with respect to u to obtain the pdf of U: f U (u).

7 6 Distributions of Functions of Random Variables 7 Example (Univariate r.v.) Let Y = yield (in tons) of pure sugar per day. Suppose Y has the pdf: f Y (y) = 2y, 0 y 1. Suppose the daily overhead=100, and the price is $300 per ton of pure sugar. Daily profit: U = 3Y 1. Find the distribution of U (i.e. find f U (u)).

8 6 Distributions of Functions of Random Variables 8 Example (Bivariate r.v.) Y 1 = proportion of gas at the beginning of a week; Y 2 = proportion of gas at the end of a week. Suppose the joint pdf is f(y 1, y 2 ) = 3y 1, 0 y 2 y 1 1. Let U = Y 1 Y 2. Determine the pdf of U.

9 6 Distributions of Functions of Random Variables 9 Distribution of the Square of a r.v. Theorem 1 Let Y be a r.v. with pdf f Y (y) and let U = Y 2. Then Proof: f U (u) = 1 2 u { fy ( u) + f Y ( u) }, 0 < u < +.

10 6 Distributions of Functions of Random Variables 10 Example Let Z N(0, 1), and U = Z 2. Show that U χ 2 (1).

11 6 Distributions of Functions of Random Variables Method of Transformations Method of Transformations Let Y be a r.v. with pdf f Y (y). Let h(y ) be a (strictly) increasing or decreasing function of Y, over the range of Y. A function h(y) is strictly increasing if: y 1 < y 2 implies h(y 1 ) < h(y 2 ). Objective: find the pdf of the r.v. U = h(y ).

12 6 Distributions of Functions of Random Variables 12 For a strictly increasing function h(y), the pdf of U = h(y ) is f U (u) = f Y { h 1 (u) } dy du. For a strictly decreasing function h(y), the pdf of U = h(y ) is f U (u) = f Y { h 1 (u) } ( dy du ). Then for h(y) strictly increasing or decreasing: { f U (u) = f Y h 1 (u) } dy. du

13 6 Distributions of Functions of Random Variables 13 Example Let Y be a rv with pdf: f Y (y) = 3y 2, 0 y 1. Let U = h(y ) = Y 2. Derive the pdf of U.

14 6 Distributions of Functions of Random Variables 14 Example Let Y be a rv with pdf: f Y (y) = 3y 2, 0 y 1. Let U = h(y ) = 2Y 2 1. Derive the pdf of U.

15 6 Distributions of Functions of Random Variables 15 Example Let Y beta(2, 2) with pdf: f Y (y) = y(1 y), 0 y 1. Let U = h(y ) = θ 1 + (θ 2 θ 1 )Y, where θ 1 < θ 2 are parameters. Derive the pdf of U.

16 6 Distributions of Functions of Random Variables 16 Example Let Y N(µ, σ 2 ) with pdf: f Y (y) = 1 σ 2 2π e (y µ) /(2σ 2), < y < +. Let U = h(y ) = (Y µ)/σ. Derive the pdf of U.

17 6 Distributions of Functions of Random Variables Method of Moment Generating Functions Uniqueness Theorem: Suppose rvs X and Y have mgfs m X (t) and m Y (t). If m X (t) = m Y (t) for all t, then X and Y have the same distribution. mgf of sums of independent random variables Let Y 1, Y 2,, Y n be independent rvs with mgfs m Y1 (t), m Y2 (t),, m Yn (t), and let U = Y 1 + Y Y n. Then m U (t) = m Y1 (t) m Y2 (t) m Yn (t).

18 6 Distributions of Functions of Random Variables 18 Example Let Y 1, Y 2 be independent r.v.s with pdfs: f Yi (y i ) = 1 β e y i/β, 0 y i < +, i = 1, 2. Find the distribution of U = Y 1 + Y 2. Recall that Y i Exp(β), and m Yi (t) = (1 βt) 1.

19 6 Distributions of Functions of Random Variables 19 Example Let Y 1, Y 2,, Y n be independent N(µ, σ 2 ) rvs. Find the distribution of Ȳ = 1/n(Y Y n ). Recall that for Y N(µ, σ 2 ), m Y (t) = exp(µt + t 2 σ 2 /2).

20 6 Distributions of Functions of Random Variables 20 Summary Method of distribution function: for continuous r.v., general Method of transformation: for continuous r.v., when h(x) is a strictly increasing/decreasing function of X Method of MGF: for both discrete and continuous. Convenient for calculating the distribution of a linear combination of independent random variables.

21 6 Distributions of Functions of Random Variables 21 Final Review Chapter 1: some basic concepts (population, sample, mean, median, variance, empirical rule) Chapter 2: theory of probability Sample-point method The Equally-likely Outcomes Model and Counting Rules Conditional Probability and Independence Laws of probability (multiplicative, additive, complementary) Law of total probability and Bayes rule Chapter 3: discrete distribution Probability mass function (PMF), cumulative distribution function CDF Expectation and variance Common distributions

22 6 Distributions of Functions of Random Variables 22 Binomial: number of successes among n trials Geometric: the trial number on which the first success occurs Hypergeometric: number of successes in a sample of size n drawn from the population of size N, among which r are successes Negative Binomial: the trial number on which the rth success occurs Poisson: the number of occurrence within a period of time/space Moment generating function Chapter 4: continuous distribution Probability density function (PDF), and CDF Expectation and variance Expectation and variance of linear functions of a random

23 6 Distributions of Functions of Random Variables 23 variable Common distributions and their properties (mean, variance, CDF, MGF etc) Uniform U(θ 1, θ 2 ) Normal N(µ, σ 2 ): some calculations related to Normal distribution Gamma(α, β) Beta(α, β) Moment generating function Chapter 5: multivariate distribution Joint pmf/pdf; joint CDF Joint distribution = Marginal distribution = marginal CDF, mean, variance of single random variables Conditional distribution (conditional pdf/pmf) = conditional probability eg P (X > 1/2 Y = 3/4); conditional expectation

24 6 Distributions of Functions of Random Variables 24 eg E(Y 2 Y 1 = 3/4) when Y 2 Y 1 U(0, Y 1 ). Independence Expectation of functions of bivariate random variables Covariance Expectation and variance of linear functions of multivariate random variables Chapter 6: derive the distribution of U = h(y 1, Y 2 ) or U = h(y ). For discrete random variables Method of distribution function Method of transformation Method of moment generating function