Probability Distributions Random Variables: Finite and Continuous A reviewFinite Random VariablesProbability DistributionsProbability Mass FunctionCumulative Distribution FunctionSlide 6Binomial Random VariableBINOMDISTExpected ValueContinuous Random VariableProbability Density Function (p.d.f)Slide 12Probability Density FunctionSlide 14Slide 15Fundamental Theorem of Calculus (FTC)Example 7 from Course FilesSlide 18Example 8 from the Course FilesExponential DistributionContinuous R.V. with exponential distributionUniform DistributionContinuous R.V. with uniform distributionFocus on the ProjectIdentify Random VariablesWhat should you do?Probability DistributionsRandom Variables: Finite and ContinuousA reviewMAT174, Spring 2004Finite Random VariablesWe want to associate probabilities with the values that the random variable takes on.There are two types of functions that allow us to do this:Probability Mass Functions (p.m.f)Cumulative Distribution Functions (c.d.f)Probability DistributionsThe pattern of probabilities for a random variable is called its probability distribution.In the case of a finite random variable we call this the probability mass function (p.m.f.), fx(x) where fx(x) = P( X = x )1all x( ) 1. Thus, 0 ( ) 1 for any value of and ( ) 1ni XiXP X x f x xf x   Probability Mass FunctionThis is a p.m.f which is a histogram representing the probabilitiesThe bars are centered above the values of the random variableThe heights of the bars are equal to the corresponding probabilities (when the width of your rectangles is 1)00.10.20.30.40.50 1 2P(X=x)Cumulative Distribution FunctionThe same probability information is often given in a different form, called the cumulative distribution function (c.d.f) or FXFX(x) = P(X ≤ x)0 ≤ FX(x) ≤ 1, for all xIn the finite case, the graph of a c.d.f. should look like a step function, where the maximum is 1 and the minimum is 0.Cumulative Distribution FunctionCumulative Distrib ution Function0.00.20.40.60.81.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14xFX(x )Binomial Random VariableLet X stand for the number of successes in n Bernoulli Trials where X is called a Binomial Random VariableBinomial Setting:1. You have n repeated trials of an experiment 2. On a single trial, there are only two possible outcomes3. The probability of success is the same from trial to trial4. The outcome of each trial is independentExpected Value of a Binomial R.V is represented by E(X)=n*pBINOMDISTBINOMDIST is a built-in Excel function that gives values for the p.m.f and c.d.f of any binomial random variableIt is located under Statistical in the Function menu–BINOMDIST(x, n, p, false) = P(X=x)–BINOMDIST(x, n, p, true) = P(X ≤ x)Expected ValueThis is average value of X (what happens on average in infinitely many repeated trials of the underlying experiment–It is denoted by XFor a Binomial Random Variable, E(X)=n*p, where n is the the number of independent trials and p is the probability of success xXxfxXEall)()(Continuous Random VariableContinuous random variables take on values in an interval; you cannot list all the possible valuesExamples: 1. Let X be a randomly selected number between 0 and 12. Let R be a future value of a weekly ratio of closing prices for IBM stock3. Let W be the exact weight of a randomly selected studentYou can only calculate probabilities associated with interval values of X. You cannot calculate P(X=x); however we can still look at its c.d.f, FX(x).Probability Density Function (p.d.f)Represented by fx(x)–fx(x) is the height of the function fx(x) at an input of x–This function does not give probabilitiesFor any continuous random variable, X, P(X=a)=0 for every number a. Look at probabilities associated with X taking on an interval of values–P(a ≤ X ≤ b)Probability Density Function (p.d.f)To find P(a ≤ X ≤ b), we need to look at the portion of the graph that corresponds to this interval.How can we relate this to integration?Aa bfXProbability Density Function( ) ( ) ( )( )( )( ).X XA F b F a P a X bP a X bP a X bP a X b          Cumulative Distribution FunctionCDF --–FX(x)=P(X ≤ x)–0 ≤ FX(x) ≤ 1, for all xNOTE: Regardless of whether the random variable is finite or continuous, the cdf, FX, has the same interpretation–I.e., FX(x)=P(X ≤ x)Cumulative Distribution FunctionFor the finite case, our c.d.f graph was a step functionFor the continuous case, our c.d.f. graph will be a continuous graphCumulative Distribution Function0.00.20.40.60.81.01.2-1 0 1 2 3tFT(t)Fundamental Theorem of Calculus (FTC)Given that –Differentiate both sides and what happens?Well, from the previous slide we can see that–If we differentiate both sides, we get that What does this say?How can we verify this claim? dxxgxG )()( dxxfxFXX)()()()('xfxFXXExample 7 from Course FilesDefine the following function:–What are the possible values of X?–Set up an integral that would give you the following probabilities:P(X < 0.5)P(X > 0.6)P(0.1 ≤ X ≤ 0.9)P(0.1 ≤ X ≤ 5)–Verify that the function is a density function –What is E(X)?elsewhere 010 if 155.37305.7)(234xxxxxxfXExpected ValueFor a finite random variable, we summed over all possible values of xFor a continuous random variable, we want to integrate over all possible values of xThis implies that dxxfxXEXX)()(Example 8 from the Course FilesLet T be the amount of time between consecutive computer crashes and has the following p.d.f. and c.d.f.–What type of r.v. is T?–Calculate P(1 < T < 5) in two different ways.–What is E(X)?0 tif 10 if 0)(0 tif 8.1610 if 0)(8.168.16tTtTettFettfExponential DistributionExponential random variables usually describe the waiting time between consecutive events.In general, the p.d.f and c.d.f for an exponential random variable X is given as follows:Any EXPONENTIAL random variable X, with parameter , has How can we verify this?xexxFxX0if10if0)(/xexxfxX0if10if0)(/( )E XContinuous R.V. with exponential distribution Probability Density Function0.00.10.20.30.40.50.6-3 0 3 6 9 12 15xfX(x )Cumulative Distribution