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Expectation and Variance: Continuous Random Variables May 08, 2006 Continuous Random Variables Expected Value Denition. Let X be a real-valued random variable with density function f (x). The expected value µ = E(X) is dened by +∞ µ = E(X) = xf (x) dx , −∞ provided the integral +∞ |x|f (x) dx −∞ is nite. Continuous Random Variables 1 Properties • If X and Y are real-valued random variables and c is any constant, then E(X + Y ) = E(X) + E(Y ) , E(cX) = cE(X) . • More generally, if X1, X2, . . . , Xn are n real-valued random variables, and c1, c2, . . . , cn are n constants, then E(c1X1+c2X2+· · ·+cnXn) = c1E(X1)+c2E(X2)+· · ·+cnE(Xn) . Continuous Random Variables 2 Example • Suppose Mr. and Mrs. Lockhorn agree to meet at the Hanover Inn between 5:00 and 6:00 P.M. on Tuesday. • Suppose each arrives at a time between 5:00 and 6:00 chosen at random with uniform probability. • Let Z be the random variable which describes the length of time that the rst to arrive has to wait for the other. • What is E(Z)? Continuous Random Variables 3 Expectation of a Function of a Random Variable Theorem. If X is a real-valued random variable and if φ : R → R is a continuous real-valued function, then +∞ E(φ(X)) = φ(x)fX (x) dx , −∞ provided the integral exists. Continuous Random Variables 4 Expectation of the Product of Two Random Variables Theorem. Let X and Y be independent real-valued continuous random variables with nite expected values. Then we have E(XY ) = E(X)E(Y ) . Continuous Random Variables 5 Example • Let Z = (X, Y ) be a point chosen at random in the unit square. • What is E(X 2Y 2)? Continuous Random Variables 6 Variance Denition. Let X be a real-valued random variable with density function f (x). The variance σ 2 = V (X) is dened by σ 2 = V (X) = E((X − µ)2) . Continuous Random Variables 7 Computation Theorem. If X is a real-valued random variable with E(X) = µ, then ∞ 2 σ = (x − µ)2f (x) dx . −∞ Continuous Random Variables 8 Properties of the variance • If X is a real-valued random variable dened on Ω and c is any constant, then V (cX) = c2V (X) , V (X + c) = V (X) . Continuous Random Variables 9 • If X is a real-valued random variable with E(X) = µ, then V (X) = E(X 2) − µ2 . Continuous Random Variables 10 • If X and Y are independent real-valued random variables on Ω, then V (X + Y ) = V (X) + V (Y ) . Continuous Random Variables 11 Example • Let X be an exponentially distributed random variable with parameter λ. • Then the density function of X is fX (x) = λe−λx . • What is E(X) and V (X)? Continuous Random Variables 12 Normal Density • Let Z be a standard normal random variable with density function 1 −x2/2 fZ (x) = √ e . 2π • What us E(X) and V (X)? Continuous Random Variables 13 Cauchy Density • Let X be a continuous random variable with the Cauchy density function a 1 fX (x) = . π a2 + x2 • What is E(X) and V (X)? Continuous Random Variables 14 Independent Trials Theorem. If X1, X2, . . . , Xn is an independent trials process of real-valued random variables, with E(Xi) = µ and V (Xi) = σ 2, and if Sn = X1 + X2 + · · · + Xn , Sn An = , n then E(Sn) = nµ , E(An) = µ , V (Sn) = nσ 2 , Continuous Random Variables 15 σ2 V (An) = . n It follows that if we set ∗ Sn − nµ Sn = √ , nσ 2 then ∗ E(Sn) = 0 , ∗ V (Sn) = 1 . ∗ We say that Sn is a standardized version of Sn Continuous Random Variables 16

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probability theory, non-technical introduction, Tamas Rudas, social sciences, precise descriptions, SAGE Publications, continuous distributions, Tamás Rudas, Stock Image, Book Description

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posted: | 5/21/2011 |

language: | English |

pages: | 17 |

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