Discrete & Continuous Random Variables - Statistics

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Expectation and Rational BehaviorThe expected value EX plays a very important role in many walks of life:casino, lottery, insurance, credit card companies, merchants, Social Security.Each time you have to ask what is the fair value of a payout X?Everybody is trying to make money or cover the cost of running the business.Read the very entertaining text discussion on pp. 96-99.It discusses psychological aspects of certain expected value propositions.15
Expectation of ϕ(X)If X is a random variable with pmf f (x), so is Y = ϕ(X) with some pmf g(y).We could find its expectation by first finding the pmf g(y) of Y and thenEY =∑ yg(y) = ∑yg(y)y∈Y (S) yBut we have a more direct formula that avoids the intermediate step of finding g(y):EY = Eϕ(X) =∑ ϕ(x) f (x) = ∑ϕ(x) f (x)x∈X(S)xThe proof is on the next slide.16
Page 1 and 2: Elements of Statistical MethodsDisc
Page 4 and 5: Bernoulli TrialsAn experiment is ca
Page 6 and 7: Application of the Geometric Distri
Page 8 and 9: Hypergeometric DistributionIn spite
Page 10 and 11: A Hypergeometric ApplicationIn a fa
Page 12 and 13: Allowing for More MissesThe experim
Page 14 and 15: The Fair Value of the PotWe can ign
Page 18 and 19: Eϕ(X) = ∑ x ϕ(x) f (x)∑ϕ(x)
Page 20 and 21: Properties of ExpectationIn the fol
Page 22 and 23: Proof for E[X 1 + X 2 ] = EX 1 + EX
Page 24 and 25: Mean Balancing Actp imoment arm x i
Page 26 and 27: The Variance of a Bernoulli Random
Page 28 and 29: Example Variance CalculationSuppose
Page 30 and 31: Var (X 1 + X 2 )If Y 1 , Y 2 are in
Page 32 and 33: Binomial DistributionThe binomial r
Page 34 and 35: Example CalculationsWhen we roll a
Page 36 and 37: Continuous Random VariablesSuppose
Page 38 and 39: Do We Have a Resolution?We started
Page 40 and 41: CDF of Uniform [0,1]F(x)0.0 0.5 1.0
Page 42 and 43: Density of Uniform[0,1]f(x)0.0 0.5
Page 44 and 45: General Density (PDF)f(x)0.0 0.1 0.
Page 46 and 47: Expectation in the Continuous CaseT
Page 48 and 49: The Variance in the Continuous Case
Page 50 and 51: Example: Engine ShutdownAssume that
Page 52 and 53: Example: Sprinkler System FailureA
Page 54 and 55: Example: Sprinkler System Failure (
Page 56 and 57: Bus Waiting Time IllustrationX 2 =
Page 58 and 59: f(x)0.00 0.01 0.02 0.03 0.04Normal
Page 60 and 61: Using pnorm in RIntroductory statis
Page 62 and 63: Normal Sampling DistributionsSevera
Page 64 and 65: Properties of the Chi-Squared Distr
Page 66 and 67:
Hole Centers(X 1 , X 2 )●(µ 1 ,
Page 68 and 69:
Student’s t DistributionWhen Z
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Discrete & Continuous Random Variables - Statistics

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