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This 20 page Class Notes was uploaded by Ms. Ari Lesch on Saturday September 26, 2015. The Class Notes belongs to ECON 671 at Iowa State University taught by Staff in Fall. Since its upload, it has received 33 views. For similar materials see /class/214442/econ-671-iowa-state-university in Economcs at Iowa State University.
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Date Created: 09/26/15
Random Variables Helle Bunzel Overview We will go over most of the general concepts Our coverage will be mostly without proofs Outline 1 Review of univariate random variables their distributions and the properties of those distributions 2 Review of multidimensional random variables and their properties 3 Finally we ll look at distributions of quadratic forms Random Variables page a m mu Random Variables Definition 1 Experiment Any activity for which the nal outcome cannot be speci ed but a set containing all possible outcome can be identi ed Definition 2 Sample Space S The set of all possible outcomes Rolling a die is an experiment What is the sample space Definition 3 An event Subset of the sample space Choose subsets A 123 and B 456 ltgt If we get 5 we say that event B has occurred Not all sample spaces are numbers Examples ltgt Usually we convert the sample space to spaces on the real line ltgt We may also be interested in real functions of the outcomes Random Variables pagEA Er mu Random Variables Definition 4 If X S gt R is a real valued function having the elements of S as its domain then X is called a random variable There are two main types of random variables ltgt A random variable is discrete if the set of outcomes is either finite or countany finite ltgtA random variable is continuous if the set of outcomes is infinitely divisible Finally PX is a probability function defined such that it assigns a prob ability to all possible events Probabilities can be assigned to different outcomes Data Outcome of a random variable P X x is the probability that the random variable X takes on the value x Random Variables page 5 r mu Density Fuctions Define a function f x from S to R that satisfies 1 For discrete distributions 0 g f x g 1 and for continuous 0 g f x 2 For discrete distributions Exf x 1 and for continuous ff x dx 1 For a discrete random variable f x P X x ltgt Note that if x is not a possible outcome not in S then fx 0 For continuous distributions P X x 0 but P A g X g B B fA f dx ltgt NOTE f x CANNOT be interpreted as point probability Random Variables page 6 r mu Example A trucking company has observed that accidents are equally likely along a certain 10 mile stretch of highway ltgt Here it makes no sense to talk about having an accident at any specific point If an accident is to happen on that stretch what is the probability of having an accident between two points a and b on that stretch ltgt PX A Z bf a That means that the density function must satisfy 1 Afxdxb10a forallOga 17310 For the next step we need Random Variables page 7 r mu Example Theorem 5 Fundamental Theorem of Calculus a b a mm m a b Ea fxgtdx fagt abfxdx ltgt fb 01 Use this This implies that 01 x E 0 10 f x 0 otherwise 0391 391x 010 What is the probability that an accident occurs on the first half of the stretch of highway Random Variables page E r mu Example What is the probability of A 12 U 79 What is the probability of B 02 U 78 U 910 Note Probability 0 vs Impossibility Random Variables page a r mu Probability Denisity Functions Let us verify whether the following functions are valid pdfs 1f X 102 x 2 f x 033C 07136 1 01 x 3 f x x2 1111 x 4 f g x2 1 111 Random Variables pagemurmn Cumulative Distribution Functions The cumulative probability function F x is defined as P X g x ltgt For the discrete this is F x Zygxf y 29 P X y ltgt For the continuous it is F x ffoof y dy gtlt Note that this definition implies that f x F x The properties of F are as follows 10 g F x g 1 2 ifx ZythenFx 2 Fy 3 F 00 1 4 F 00 0 5PangbFb Fa Drawings Random Variables page Erma Multivariate Random Variables A multivariate random variable X can be written as X1 X 32 Xn where X i 1 n are random variables Examples of multivariate random variables ltgt Multiple characteristics gtlt Cars gtlt People The density of X is denoted by f x1x2 xn Discrete If all the individual rvs are discrete Continuous If all the individual rvs are continuous Random Variables pagelZUflElEl Multivariate Random Variables Definition off x1x2 xn p A 39 39Zx1x2xnEAf 351le quot13511 f 39 39 39fx1x2xneAfx1X2Xn dx1dx2dxn For the discrete case EarnquotImagnf x1x2 xn 1 For the continuous case f oo f oof x1x2 xn dxldxzdxn 1 Random Variables pagei n Example A pair of dice two colors Red green X1 Number of eyes on red X2 Sum of eyes on red and green Elements in S What is the density of this stochastic variable What is the probability that red is two or less and the sum is 5 or less 2 5 PM 2 Z fX1X2 x121 x2x11 Random Variables pagetzlnft Example Big screen TV production TVs 3 x 4 feet Coating machine produces a flaw Each point on the screen is equally likely to recieve the flaw Label the area 1515 x 2 2 What is the density of flaws What is the probability that the flaw ends in an area with width W and height H We are looking for the function where d b C A fltxygtdxdyW forall 2 altb 2and 15 cltd 15 Random Variables pageisurmn Example Differentiate wrt d 6 b a d c 6 d b 1 b a Differentiate wrt b a b 1 617 61 fa fltxdgtdx fltbdgt E 1Z 30 f 3511 Z 1 121xyE 22gtlt 1515 Random Variables pageisurmn Multivariate CDFs When X is discrete the multivariate CDF is definied as F b1bn P Xi g bi i 1n Z 2 f x1xn x1gb1 x11ng When X is continuous the multivariate CDF is definied as 12 121 F b1bn p X1 3 bi i 1n m f x1xn Properties of the multivariate CDF 1 limbigtooF b1bn 0139 1n 2 Iimbi oolizlwvn F 171 l7 1 3Fa lt Fb fora lt 17 Random Variables page urmn Multivariate CDFs In the continuous case we can move from CDF to pdf much as before an Theorem 6 f MINx ax1maan x1 753852er IS continuous In the discrete case it is much more complicated Two dimensional example ltgt X Y binary discrete variable with joint cumulative distribution function FX Y ltgt x1 lt x2 lt and y1 lt 12 lt possible outcomes of X and Y F Xlzyl f 361 Fx1yj Fx11j 1zj 2 2 FXi1y1 FXi 1y1zi2 2 F xiii1 F xi1 F xi 1m F xi 14H Random Variables pagel n Example Continuing TV screen example Density was f X11362 1 22 X11 1515 X2 Calculate CDF b1 172 1 F0711 172 2 oo oo EH42 x11 1515 X2dxzdx1 1 171 12 2 E 00 1i 2r2 x1 00 1 1515 X2dxzdx1 Random Variables pageiaurmn Example 2 15 b2 00 1 1515 x2 dxz 115oo 172 1515130 1 1515 172 15151352 3 39115oo 172 172 15gt1 1515 b2 1 b1 172 F071 172 ELwH zz X1001 1515 X2dX2dX1 1 171 115oo b2001 22 96061361 1724 15 171 12 1 1515b2001 22 36061361 b1 b1 2 00 1W x1 61361 2 1W M 2 1dx1 1pm b1 21de 1 22 171 171 2 43912oo 171 Random Variables page 2n m mu Example 1 F M 115ltb2gt 1m b1 m 2gt 412 w 172 15 12 11515ltb2gt 1W b1 h 2 412 w 17 2 14 gt11500 122 1500 H2100 b 15 b 2 14515 b2 1 2r2 b1 17 15 2 3 1 1515 b212oo b1 Random Variables pagem mun
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