Random Variables and Probability DistributionsRandom VariablesSlide 3Slide 4Slide 5Random Variable EventsProbability DistributionsProbability Mass FunctionSlide 9Probability Density FunctionSlide 11Slide 12Cumulative Distribution FunctionSlide 14Slide 15Slide 16Slide 17Joint Probability DistributionsMarginal Probability DistributionsDiscrete Marginal DistributionsContinuous Marginal DistributionsConditional DistributionsSlide 23Slide 24Slide 25Slide 26Slide 27IndependenceSlide 29Slide 301Random Variables andProbability DistributionsX is a random variable (r.v.)Probability Distribution•Discrete probability mass function (p.m.f.)•Continuous probability density function (p.d.f.)•Cumulative distribution function (c.d.f.)Joint Probability Distribution•Joint p.m.f or p.d.f.•Marginal distributions•Conditional distributions, Independence2Random VariablesX is a mapping from each simple event in the sample space S to a real number•Discrete: finite or countable set of values•Continuous: uncountable set of values3Random VariablesDiscrete random variable: only takes on certain values in a finite or countable set.•Flip two coins: X = # of heads HH  X = 2 HT  X = 1 TH  X = 1 TT  X = 0•Roll a die: X = # of dots on the side facing up•Color: X = # of students in a class whose favorite color is red•Post office: X = # of people in line4Random VariablesContinuous random variable: takes on any values within some range (i.e., infinitely many values in an uncountable set).•X = the lifetime (hours) of a light bulb•X = the weight of the next package that you take to the post office•X = the length of time to play 18 holes of golf•X = the annual income of Texas residents5Machine Breakdown Example S = {electrical, mechanical, misuse}Let X = the repair cost associated with a failureSample SpaceRandom Variable( Repair Cost )MisuseElectricalMechanicalReal Number Line50 200 350Random Variables6Random Variable EventsFlip two coins: X = # of heads•Exactly one head: [X = 1]•No more than one head: [X  1]Post office: X = # of people in line•At least 3 and fewer than 10 people: [3  X < 10]•More than 3 and at most 10 people: [3 < X  10]Light bulb: X = lifetime of the bulb•More than 50 hours: [X > 50]•Never turns on: [X = 0]7Probability DistributionsThe probabilities assigned to the values of XA discrete r.v. X has probability mass function (p.m.f.): fX(x) = P[X = x]A continuous r.v. X has probability density function (p.d.f.): fX(x)8Probability Mass FunctionfX(x) maps the possible values of the discrete r.v. X to probabilities on the interval [0,1].fX(x) = P[X = x] = probability that X = x, where x is from a finite or countable set.p.m.f. values are probabilities.Properties of the p.m.f.:1) 0  fX(x)  1 for all x2) x fX(x) = 19Probability Mass FunctionExample: Flip two coins•S = { HH, HT, TH, TT }•X = # of headsfX(0) = P[X = 0] = P(TT) = 1/4fX(1) = P[X = 1] = P(HT or TH) = 1/4 + 1/4 = 1/2fX(2) = P[X = 2] = P(HH) = 1/40 < fX(x) < 1 for x = 0, 1, 2; o/w fX(x) = 0x fX(x) = fX(0) + fX(1) + fX(2) = 110Probability Density FunctionfX(x) describes the distribution of values of X over a continuous range.p.d.f. values are NOT probabilities.Probabilities are found by calculating the area under the fX(x) curve.baXdxxfbXaP)(][11Probability Density FunctionProperties of the p.d.f.:1) fX(x)  0 for all x2)The probability of a single point is zeroIf X is a continuous r.v., then for any a and b,P[a  X  b] = P[a < X  b] = P[a  X < b] = P[a < X < b]1)( dxxfX0)(][ dxxfaXPaaX12Probability Density FunctionX = lifetime (hrs) of a certain kind of radio tubeP[X < 150] Note: 31100100150100100150100x0)( o/w ;100for 100)(2 xfxxxfXX110010001001001001002xdxxdxxdxxfX1501002150100)(13Cumulative Distribution FunctionFX(x) = P[X  x] •Discrete r.v. X: for  < x < •Continuous r.v. X: for  < x < c.d.f. values are probabilitiesxXXdttfxF )()(xtXXtfxF )()(14Cumulative Distribution FunctionProperties of the c.d.f.:1) 0  FX(x)  1 for all x2) If x  y, then FX(x)  FX(y)15Cumulative Distribution FunctionDiscrete Example: Flip 2 coins•FX(0) = P[X  0] = fX(0) = 1/4•FX(1) = P[X  1] = fX(0) + fX(1) = 1/4 + 1/2 = 3/4•FX(2) = P[X  2] = fX(0) + fX(1) + fX(2) = 1 FX(x) = 0 for x < 0 FX(x) = 1/4 for 0  x < 1 FX(x) = 3/4 for 1  x < 2 FX(x) = 1 for x ≥ 216Machine Breakdown Example•X = the repair cost associated with a failure•P[X = 50] = 0.3, P[X = 200] = 0.2, P[X = 350] = 0.5CDF for XCumulative Distribution Function50 200 350Repair cost0.30.5117Cumulative Distribution FunctionContinuous Example: Lifetime of radio tube•For x > 100 FX(x) = 0 for x  100 FX(x) = for x > 100xxtdttdttfxFxxxXX1001100100100100100)()(1001002x100118Joint Probability DistributionsBivariate Case has two r.v.’s X and Y•fX,Y (x, y) is the joint p.m.f. or joint p.d.f.Joint p.m.f.: fX,Y (x, y) = P[X = x, Y = y]•0  fX,Y (x, y)  1 for all x, yx y fX,Y (x, y) = 1Joint p.d.f.:•fX,Y (x, y)  0 for all x, y•  1),(,dydxyxfYXdydxyxfAYXPAYX ),(]),[(,19Marginal Probability DistributionsFor discrete r.v.’s X and Y fX (x) = y fX,Y (x, y) fY (y) = x fX,Y (x, y) For continuous r.v.’s X and Y dxyxfyfdyyxfxfYXYYXX),()(),()(,,20Discrete Marginal DistributionsExample:21Continuous Marginal DistributionsExample: 0),( o/w ;0,0for 2),(,2,yxfyxeeyxfYXyxYX    xxyxyxYXXeeeeeedyeedyyxfxf00202,222),()(    yyxyxyYXYeeeeeedxeedxyxfyf2020202,22122),()(22Conditional DistributionsGiven fX,Y (x, y), fX (x) > 0, fY (y) > 0, we can calculate•fX|Y (x | y) = conditional distribution of X given Y = y•fY|X (y | x) = conditional distribution of Y given X = x)(),()|()(),()|(,|,|xfyxfxyfyfyxfyxfXYXXYYYXYX23Conditional DistributionsFor