Understanding Expected Values: A Guide to Probabilities and

School

North Carolina State University**We aren't endorsed by this school

Course

ST 371

Subject

Computer Science

Date

Dec 12, 2024

Pages

Uploaded by AdmiralExploration4857

Lecture 9: Expected Values

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsIntroduction▶The last lecture we discussed how we can computeprobabilities based on distributions that arerepresentative of eliminating the assumption of equallylikely outcomes.▶Since dropping the equally likely outcomes assumptioncreates a distribution of probability among theoutcomes, or values the random variables, we becomeinterested in certain properties that these randomvariables possess.▶Namely, we are interested in the average value therandom variable attains and the variance between thevalues of the random variable.▶We will use knowledge of the probability distributions ofthese random variables to find answers to thesequestions.

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsAverage Value of a Random Variable▶Suppose we are interested in the average number ofcourses that a student is taking here at NCSU thissemester. Further suppose thatXis the number ofcourses a randomly selected student is taking and out ofthe 35000 students at this university we have thefollowing distribution of students per number of courses:x123456Registered1050105045508750136505950▶How might we go about computing the average numberof courses per student?

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsExpected ValuesDefinitionLetXbe a discrete random variable with set of possiblevaluesDand pmfp(x). Theexpected valueofX, denotedbyE(X)orµXis given byE(X) =Xx∈Dxp(x)▶In the creation of the above formula we saw how we canuse a specific probability distribution to compute theaverage value of a random variable. We can do thesame thing for when a pmf is a function with aparameter instead of a specific number.

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsExample 1Exercise1.LetXbe a Bernoulli distributed random variable. Whatis the expected value ofX?2.LetYbe a geometric distribution with parameterp.What is the expected value ofY?

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsExpected Value of a Function ofX▶Often times we are interested in computing theexpected value of afunctionof the random variablerather than the random variable itself.▶For example, what if we are interested in the averagenumber of credit hours that a student is taking, ratherthan the number of courses.▶Assume that all courses are 3 credit hours then if we letYequal the number of credit hours we can easily seethis as a function ofX.

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsExpected Value of a Function ofXDefinitionIf the random variableXhas a set of possible valuesDandpmfp(x), then the expected value of any functionh(X),denoted byE(h(X))orµh(X)is computed byE(h(X)) =Xx∈Dh(x)p(x)

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsSpecial Case▶Often times the functionh(X)of the random variableXthat we are interested is a special type called alinearfunction.▶This means that the function has the formh(X) =aX+bfor some constantsaandb. Then wecan find the expected value of a function with this formbyDefinitionE(aX+b) =aE(X) +b▶Proof:

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsVariance of a Random VariableX▶Also of interest in just how spread out those values are.Since we could use the probability distribution tocompute a measure of center, so can we use theprobability distribution to compute a measure ofvariance.▶This can be done through considering the expectation ofthe function ofXthat has the formh(X) = (X−µ)2.DefinitionLetXhave pmfp(x)and expected valueµ. Then thevarianceofX, denoted asV ar(X)orσ2XisV ar(X) =E((X−µ)2) =Xx∈D(x−µ)2p(x)

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsShortcut formula forV ar(X)▶The formula given for computing the variance ofXcaninvolve numerous arithmetic computations. We can finda more computationally friendly formula by expandingthe binomial:PropositionV ar(X) =E(X2)−[E(X)]2▶Proof:

Prob & Stats:Lecture 9IntroductionExpected ValueMotivationDefinitionExampleExpectation of a function ofXVarianceDefinitionExampleComputational FormulaExampleLinear FunctionsVariance of a Linear Function▶Just like we could be interested in computing theexpected value of a function of a random variable wemight also be interested in the variance of that function.▶V ar(h(X)) =∑x∈D[h(x)−E(h(X))]2p(x)▶Ifh(X)is a linear function then we have:PropositionV ar(aX+b) =a2σ2X▶Proof: