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# 503 Class Note for STAT 416 at PSU

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This 12 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 24 views.

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Date Created: 02/06/15

Joint Distribution 0 We may be interested in probability statements of sev eral RVs 0 Example Two people A and B both ip coin twice X number of heads obtained by A Y number of heads obtained by B Find PX gt Y 0 Discrete case Joint probability mass function pz y PX 51 Y y Two coins one fair the other twoheaded A ran domly chooses one and B takes the other X i l A gets head Y i l B gets head i 0 A gets tail 0 B gets tail Find PX 2 Y o Marginal probability mass function of X can be ob tained from the joint probability mass function px 3 WW Z 19567 y ypfr7ygt0 Similarly p y Z WI 2 frrp7ygt0 0 Continuous case Joint probability density function f 1 y PX Y E R Rfxydxdy o Marginal pdf tk ame WZfw w 0 Joint cumulative probability distribution function of X and Y FabPX aY b ooltabltoo o Marginal cdf FXa Fa oo Fyb Foo b o Expectation E 9X Y 20 05 9x ypx y in the discrete case LOO ff 9x y f 1 ydady in the continuous case 0 Based on joint distribution we can derive EaX bY aEX bEY Extension ElaiXi G2X2 aanl G1EX1 a2EX2 0 Example EX X is binomial with n p X i l z th ip is head Z 7 0 2th ip is tail X EX ZEPQ 71 0 Assume there are n students in a class What is the expected number of months in which at least one stu dent was bom Assume equal chance of being born in any month Solution Let X be the number of months some stu dents are born Let Xi be the indicator RV for the 2th month in which some students are born Then X Hence ll EX 12EX1 1213x1 l 12 l Em Independent Random Variables o X and Y are independent if HXng HX HY o Equivalently Fa b FXaFyb 0 Discrete px y pXxpyy 0 Continuous x 3 fXfYZ 0 Proposition 23 If X and Y are independent then for function h and g EgXhY EgXEhY Covariance 0 De nition Covariance of X and Y COWX Y EKX EXY EY o CovXX EX EX2 VarX COUX Y EXY EXEY o If X and Y are independent COUX Y 0 0 Properties 1 COUX X VarX 2 COUX Y COUY X 3 COUCX Y CCOUX Y 4 COUX Y Z COUX Y COUX Z Sum of Random Variables o Ifos are independent 2 1 2 n VarZ Xi VarX Var l aZXi Zn aZZVaNXi 239l 11 0 Example Variance of Binomial RV sum of indepen dent Bernoulli RVs VarX np1 p Moment Generating Functions 0 Moment generating function of a RV X is t W Ere i Zxpxgt0 txplt Xdiscrete 7 foo 5 f 5Ud L X continuous o Moment of X the nth moment of X is E X n EX We t 0 where We is the nth order derivative 0 Example 1 Bernoulli with parameter p Mt pet l p for any t 2 Poisson with parameter A Mt amt 1 for any t 0 Property 1 Moment generation function of the sum of independent RVs sz l n are independent Z X1X2 X717 205 905 0 Property 2 Moment generating function uniquely de termines the distribution 0 Example 1 Sum of independent Binomial RVs 2 Sum of independent Poisson RVs 3 Joint distribution of the sample mean and sample variance from a normal porpulation Important Inequalities 0 Markov Inequality If X is a RV that takes only non negative values then for any a gt 0 Ele PX 2a 3 o Chebyshev s Inequality If X is a RV with mean u and variance 02 then for any value 19 gt O 2 039 PX uZ Sg 0 Examples obtaining bounds on probabilities 10 Strong Law of Large Numbers 0 Theorem 21 Strong Law of Large Numbers Let X1 X2 be a sequence of independent random variables having a common distribution Let EXZ u Then with probability 1 X1X2H39Xn n u as n gtoo 11 Central Limit Theorem 0 Theorem 22 Central Limit Theorem Let X1 X2 be a sequence of independent random variables having a common distribution Let E Xi u Va Xi 02 Then the distribution of X1X2Xn nl IxE tends to the standard normal as n gt 00 That is X1X2Xn nl P 2 0 1 Z 2 WQd cp gt 2WOoe 1 0 Example estimate probability 1 Let X be the number of times that a fair coin ipped 40 times lands heads Find PX 20 2 Suppose that orders at a restaurant are iid random variables with mean u 8 dollars and standard deviation 0 2 dollars Estimate the probability that the rst 100 customers spend a total of more than 840 Estimate the probability that the rst 100 customers spend a total of between 780 and 820 12

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