Discrete & Continuous Random Variables - Statistics

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Var (X 1 + X 2 )If Y 1 , Y 2 are independent discrete random variables, then E(Y 1 Y 2 ) = EY 1 · EY 2 .E(Y 1 Y 2 ) = ∑y1 , y 2y 1 y 2 f (y 1 ,y 2 ) = ∑y1∑y2y 1 y 2 f Y1 (y 1 ) f Y2 (y 2 )= ∑y1y 1 f Y1 (y 1 )∑y2y 2 f Y2 (y 2 ) = EY 1 · EY 2For independent discrete random variables X 1 and X 2 we haveVar (X 1 + X 2 ) = Var X 1 + Var X 2With µ 1 = EX 1 , µ 2 = EX 2 , Y 1 = X 1 − µ 1 and Y 2 = X 2 − µ 2 with EY 1 = EY 2 = 0=⇒ Var (X 1 + X 2 ) = Var (X 1 + X 2 − µ 1 − µ 2 ) = Var (Y 1 +Y 2 ) = E(Y 1 +Y 2 ) 2= E[Y1 2 + 2Y 1Y 2 +Y2 2 ] = EY 1 2 + 2E[Y 1Y 2 ] + EY22= Var Y 1 + EY 1 · EY 2 + Var Y 2 = Var Y 1 + Var Y 2= Var X 1 + Var X 229
Binomial <strong>Random</strong> <strong>Variables</strong>Let X 1 ,...,X n be independent with X i ∼ Bernoulli(p). ThenY = X 1 + ... + X n =n∑ X ii=1is called a binomial random variable and we write Y ∼ Binomial(n; p).Y = # of successes in n independent Bernoulli trials with success probability p.EY = E( n∑Var Y = Var( n∑i=1i=1X i)=X i)=n∑ EX i =i=1n∑ Var X i =i=1n∑ p = npi=1n∑ p(1 − p) = np(1 − p)i=130
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 16 and 17: Expectation and Rational BehaviorTh
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 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

Discrete & Continuous Random Variables - Statistics

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