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Stochastic CS616 Models in Computer Science Instructor: Peter Kemper R 006, phone 221-3462, email:kemper@cs.wm.edu Office hours: Mon,Wed 3-5 pm Grader: Yunlian Jiang, R 101B, email: jiang@cs.wm.edu Office hours: Tue,Thu 2-3 pm Today: Moment Generating Functions Quick Reference: Ross, Ch. 2.6 1 Overview this is the plan Probability Theory Random Variables Conditional Probability Conditional Expectation Markov Chains Exponential Distribution & Poisson Process Continuous Time Markov Chain Tools: - Mobius 2 Today s topics what it is good for Moment generating function Alternative characterization of a distribution 1-1 correspondence Certain calculations are simpler with this concept than with distributions Sums of rvs: Linear translation, Convolution of independent variables Difference and differential equations related to a stoch. process Obtain moment generating function Perform calculations Match resulting moment generating function with those of known distributions, obtain resulting distribution Probability generating function: z transform Laplace-Stieltjes transform Characteristic function: Fourier transform 3 Common procedure Special cases of moment generating function : Overview Moment generating function Examples Binomial distribution Poisson distribution Exponential distribution Normal distribution Binomial distribution Poisson distribution Exponential distribution Normal distribution Sums of independent rvs of some distributions Joint moment generating function 4 Moment Generating Function Definition: Moment Generating Function For random variable X, moment generating function (t) defined as The derivatives of (t) for t=0 are of interest: n-th derivative gives n-th moment and in general: 5 (t) of the Binomial distribution with parameters n,p First moment Second moment and variance 6 (t) of the Poisson distribution with parameters First moment Second moment and variance 7 (t) of the Exponential distribution with parameters First moment Second moment and variance 8 Properties Theorem: Linear Translation Let Y = aX + b then Proof: [Trivedi, Theorem 4.4] based on linearity of expectation. 9 (t) of the Normal distribution with parameters and 2 Z standard normal distribution X normal distribution Second moment and variance 10 Properties Theorem: Convolution [Trivedi, Theorem 4.5] Let X1,X2, ,Xn be mutually independent rvs on a given probability space and If Xi(t) exists for all i, then Y(t) exists, and Proof: based on independence of rvs. Note: This helps to find the transform of a sum of independent rvs without any n-dimensional integration. But how to recover the distribution from the resulting transform? 11 Properties Theorem: Correspondence / Uniqueness [Trivedi, Theorem 4.6] If X(t) = Y(t) for all t, then X and Y have the same distribution. Proof: omitted So moments characterize a distribution, note that 12 Example 2.43 Given X with What is P{X=0} ? Answer: Given (t) matches (t) of a Poisson random variable with =3. 1-1 one correspondence (t) and distribution So X is Poisson random variable pmf of X So 13 Two means, which is better for what ? Distribution: Focuses on probabilities of events Math for combined rvs based on summation (discrete case) or integration (continuous case) Moment generating function: Focuses on moments Based on power series of moments Math for combined rvs Particularly simple for summation of independent rvs From cdf to Calculate mgf: expectation of a function of an rv Use uniqueness and knowledge of mgfs for distributions From mgf to cdf: Let s see what X+Y turns out to be for different distributions 14 Example 2.44: Sums of Independent Binomial RVs Let X, Y be independent Binomial RVs, parameters X: (n,p) Y: (m,p) Distribution of X + Y Solution Moment generating function Note: Moment generating function of Binomial Z is So distribution of Z=X+Y is a Binomial distribution with parameters k=m+n and p 15 Example 2.45: Sums of Independent Poisson RVs Let X, Y be independent Poisson RVs, parameters X: 1 Y: 2 Distribution of X + Y Solution Moment generating function Note: Moment generating function of Poisson Z is So distribution of Z=X+Y is a Poisson distribution with parameters = 1+ 2 16 Example 2.46: Sums of Independent Normal RVs Let X, Y be independent Normal RVs, parameters X: 1, 12 Y: 2, 22 Distribution of X + Y Solution Moment generating function Note: Moment generating function of Normal Z is So distribution of Z=X+Y is a Normal distribution with parameters = 1+ 2 , 2 = 12 + 22 17 Poisson Paradigmn Says the number of successes in n trials that are either independent or at most weakly dependent is, when the trial success probabilities are all small, approximately a Poisson random variable Why ? sum of independent Bernoulli rvs with pi small for small |x|: (1+x) ex and small pi makes pi(et-1) small since the moment generating func of a sum of ind. Xi is the product of the moment generating functions of Xi matches moment generating func of Poisson rv, = p i 18 Poisson Paradigm What is new ? Trials can have different success probabilities Trials can be weakly dependent (Argumentation not given here, previous slide based on independence!) 19 Joint Moment Generating Function Example: Multivariate Normal Distribution Z1, ,Zn independent standard normal rvs For constants aij, i X1, ,Xm have a multivariate normal distribution Given X1, , Xn rvs and t1, , tn real values the joint moment generating function for X1, , Xn is Let s consider rv Y= tiXi 20 Multivariate Normal Distribution Note: sum of ind. normal rvs (here Zi) is normal rv So Xi is normal rv and tiXi is normal rv For use moment generating function of normal rv Y at t=1 and set parameter values accordingly 21 Chi-square distribution Z1, ,Zn independent standard normal rvs then rv has a chi-square distribution with n degrees of freedom Moment generating function so 22 Joint Distribution of Sample Mean & Sample Variance from a Normal Distribution Let i.i.d. Xi each with mean , variance 2 Sample mean Sample variance One can show that Let Xi have a normal distribution So has a normal distribution and also have a joint multivariate normal distribution By comparison of multivariate normal distribution with one can argue for independence of with some further argumentation we obtain 23 Proposition 2.5 Proposition Let i.i.d. Xi each with mean , variance 2 then 1) its sample mean and sample variance S2 are independent, 2) 3) is a normal rv with is a chi-squared rv with n-1 degrees of freedom. 24 Summary Moment generating function Examples Binomial distribution Poisson distribution Exponential distribution Normal distribution Binomial distribution Poisson distribution Exponential distribution Normal distribution Sums of independent rvs of some distributions Joint moment generating function Independence of sample mean and sample variance 25

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