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UA MATH 115A - Finite Random Variables

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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 VariablesA random variable is a numerical value associated with the outcome of an experiment.Examples:The number of heads that appear when flipping three coinsThe sum obtained when two fair dice are rolledIn 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 TimesSuppose 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 valuesThe values are whole numbersWhen 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.ProbabilitiesJust as with events, we can also talk about probabilities of random variables.The probability a random variable assumes a certain value is written asP(X =x)Notice that X is the random variable for the # of heads and x is the value the variable assumes.ProbabilitiesWe 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 FunctionWe 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   xXPxfXProperties of Probability Mass FunctionBecause 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 AllxXxfRepresenting 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 FunctionThe 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 #sRange: 0≤ Fx(x) ≤1Fx(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 #sRange: 0≤ Fx(x) ≤1Fx(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) = 1/8If 1≤ x <2, then Fx(x) = P(X ≤ x) = P(X=1) +P(X=0)= 1/8+3/8=4/8If 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 noticeThe graph is a step-wise function. This is typically what you will see for finite discrete random variables.Domain: the set of all real #sRange: 0≤ Fx(x) ≤1Fx(x) = P(X≤x)As x →∞, Fx(x) →1 AND As x →-∞, Fx(x) →0At 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 VariableExpected 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)( xXPxExampleBox 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 $100.00fX(X) 0.40 0.30 0.20 0.10QuestionsWhat is P(X=25)?What is P(X≤25)?What P(X≥5)?Graph the c.d.f.What is the E(X)?Answers2.0)25($)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     6.04.011111515XFXPXPXPAnswer(CDF)Cumulative Distribution Function0.00.20.40.60.81.01.2-20 0 20 40 60 80 100 120xFX(x )Expected Valuex)(xfX)(xfxXX $ 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 VariablesBernoulli Random Variables are a special case of discrete random variablesIn a Bernoulli Trial there are only two outcomes: success or failureBernoulli Random VariableLet X stand for the number of successes in n Bernoulli Trials, X is called a Binomial Random VariableBinomial Setting:1. You have n repeated trials of an experiment.


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UA MATH 115A - Finite Random Variables

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