Probability Distributions: Finite Random VariablesRandom VariablesExample—Flipping a Coin 3 TimesExample (cont)ProbabilitiesSlide 6Probability Mass FunctionProperties of Probability Mass FunctionRepresenting the p.m.f.Things to notice:Cumulative Distribution FunctionGraphing the c.d.f.Graphing (cont)Slide 14The c.d.f.The graph of the c.d.f.Things to noticeExpected Value of Finite Discrete Random VariableExampleQuestionsAnswersAnswer(CDF)Expected ValueBernoulli Random VariablesBernoulli Random VariableLoaded CoinSlide 27Loaded coinSlide 29Loaded Coin: p.m.f.Loaded Coin: Graph of pmfYour Turn!Probability Distributions: Finite Random VariablesRandom VariablesA random variable is a numerical value associated with the outcome of an experiment.Examples:The number of heads that appear when flipping three coinsThe sum obtained when two fair dice are rolledIn both examples, we are not as interested in the outcomes of the experiment as we are in knowing a number value for the experimentExample—Flipping a Coin 3 TimesSuppose that we flip a coin 3 times and record each flip.S = {HHH, HHT, HTH, HTT, THH, THT, TTT, TTT}Let X be a random variable that records the # of heads we get for flipping a coin 3 times.Example (cont)The possible values our random variable X can assume:X = x where x = {0, 1, 2, 3}Notice that the values of X are:Countable, i.e. we can list all possible valuesThe values are whole numbersWhen a random variable’s values are countable the random variable is called a finite random variable.And when the values are whole #’s the random variable is discrete.ProbabilitiesJust as with events, we can also talk about probabilities of random variables.The probability a random variable assumes a certain value is written asP(X =x)Notice that X is the random variable for the # of heads and x is the value the variable assumes.ProbabilitiesWe can list all the probabilities for our random variable in a table.The pattern of probabilities for a random variable is called its probability distribution.For a finite discrete random variable, this pattern of probabilities is called the probability mass function ( p.m.f ).X=x P(X=x)0 1/81 3/82 3/83 1/8Probability Mass FunctionWe consider this table to be a function because each value of the random variable has exactly one probability associated with it.Because of this we use function notation to say:X=x P(X=x)0 1/81 3/82 3/83 1/8 xXPxfXProperties of Probability Mass FunctionBecause the p.m.f is a function it has a domain and range like any other function you’ve seen:Domain: {all whole # values random variable}Range: Sum: 10 xfX 1 AllxXxfRepresenting the p.m.f.Because the p.m.f function uses only whole # values in its domain, we often use histograms to show pictorially the distribution of probabilities.Here is a histogram for our coin example:Things to notice:The height of each rectangle corresponds to P(X=x)The sum of all heights is equal to 1Cumulative Distribution FunctionThe same probability information is often given in a different form, called the cumulative distribution function, c.d.f.Like the p.m.f. the c.d.f. is a function which we denote as Fx(x) (upper case F) with the following properties:Domain: the set of all real #sRange: 0≤ Fx(x) ≤1Fx(x) = P(X≤x)As x →∞, Fx(x) →1 AND As x →-∞, Fx(x) →0Graphing the c.d.f.Let’s graph the c.d.f. for our coin example.According to our definitions from the previous slide:Domain: the set of all real #sRange: 0≤ Fx(x) ≤1Fx(x) = P(X≤x)Graphing (cont)Here’s the p.m.f. :Because the domain for the cdf is the set of all real numbers, any value of x that is less than zero would mean that Fx(x) is 0 since there is no way for a flip of three coins to have less than 0 heads. The probability is zero!Also, because the number of heads we can get is always at most 3, Fx(x) = 1 when x ≥ 3.X=x 0 1 2 3P(X=x) 1/8 3/8 3/8 1/8Graphing (cont)Now we need to look at what happens for the other values.If 0≤ x <1, then Fx(x) = P(X ≤ x) = P(X=0) = 1/8If 1≤ x <2, then Fx(x) = P(X ≤ x) = P(X=1) +P(X=0)= 1/8+3/8=4/8If 2≤ x <3, then Fx(x) = P(X ≤ x) = P(X=2)+P(X=1) +P(X=0)= 1/8+3/8+3/8=7/8The c.d.f.All of the previous information is best summarized with a piece-wise function: 3132872184108100xifxifxifxifxifxFXThe graph of the c.d.f.Things to noticeThe graph is a step-wise function. This is typically what you will see for finite discrete random variables.Domain: the set of all real #sRange: 0≤ Fx(x) ≤1Fx(x) = P(X≤x)As x →∞, Fx(x) →1 AND As x →-∞, Fx(x) →0At each x-value where there is a jump, the size of the jump tells us the P(X=x). Because of this, we can write a p.m.f. function from a c.d.f. function and vice-versaExpected Value of Finite Discrete Random VariableExpected Value of a Discrete Random Variable is Note, this is the sum of each of the heights of each rectangle in the p.m.f., multiplied by the respective value of X in the pmf. xall)( xXPxExampleBox contains four $1 chips, three $5 chips, two $25 chips, and one $100 chip. Let X be the denomination of a chip selected at random. The p.m.f. of X is displayed below.X $1.00 $5.00 $25.00 $100.00fX(X) 0.40 0.30 0.20 0.10QuestionsWhat is P(X=25)?What is P(X≤25)?What P(X≥5)?Graph the c.d.f.What is the E(X)?Answers2.0)25($)25$( XfXP9.02.03.04.0)25($)5($)1($)25$()5$()1$()25$(XXXfffXPXPXPXP9.0)25($)25$( XFXP6.01.02.03.0)100($)25($)5($)5$(XXXfffXP 6.04.011111515XFXPXPXPAnswer(CDF)Cumulative Distribution Function0.00.20.40.60.81.01.2-20 0 20 40 60 80 100 120xFX(x )Expected Valuex)(xfX)(xfxXX $ 1 0.4 0.40 $ 5 0.3 1.50 $ 25 0.2 5.00 $ 100 0.1 10.00Sum 1.0 $16.90Bernoulli Random VariablesBernoulli Random Variables are a special case of discrete random variablesIn a Bernoulli Trial there are only two outcomes: success or failureBernoulli Random VariableLet X stand for the number of successes in n Bernoulli Trials, X is called a Binomial Random VariableBinomial Setting:1. You have n repeated trials of an experiment.
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