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stat400lec19 - same the collection of the two random...

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Statistics 400 Random Sample Mathematical Expectation Definition: If f(x1,x2) is the p.m.f. of discrete random variables X1 and X2. Then the mathematical expectation or the expected value of function u(X1,X2) E[u(X1,X2)]= ∑∑ ) , ( ) , ( 2 1 2 1 x x f x x u The random variables X1 and X2 are independent if f(x1,x2)=f 1 (x1)f 2 (X2) And if u(X1,X2)=u1(X1)u2(X2) E[u(X1,X2)]= ∑∑ ) , ( ) , ( 2 1 2 1 x x f x x u = ∑∑ ) ( ) ( ) ( ) ( 2 2 1 1 2 2 1 1 x f x f x u x u = ) ( ) ( ) ( ) ( 2 2 2 2 1 1 1 1 x f x u x f x u =E[u(X1)]E[u(X2)] Some Rules for Means: 1. If X and Y are random variables with μ X , and μ Y, as their respective means: a. μ X+Y = μ X + μ Y b. μ X-Y = μ X – μ Y 2. If X is a random variable and a and b are constants: a. X bX a bX a b a μ μ μ μ * + = + = + In general, we have Ping Ma Lecture 19 Fall 2005 - 1 -
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Rules for Variances: 1. If X and Y are independent: a. 2 2 2 Y X Y X σ σ σ + = + b. 2 2 2 Y X Y X σ σ σ + = - 2. If X is a random variable and a and b are constants: a. 2 2 2 2 2 2 0 X X a bX a b b σ σ σ σ + = + = + In general, we have, If the distribution of the independent random variables X1 and X2 is the
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Unformatted text preview: same, the collection of the two random variables X1 and X2 is called a random sample of size 2 from the distribution. If the distribution of the independent random variables X1, X2, …,Xn is the same, the collection of the two random variables X1, X2, …,Xn is called a random sample of size n from the distribution. Ping Ma Lecture 19 Fall 2005- 2 -Example Let X1, X2 and X2 be a random sample from an exponential distribution with mean of 1. (1) What is the joint pdf of X1, X2 and X3, (2)P(0<X1<1, 2<X2<4, 3<X3<7) Ping Ma Lecture 19 Fall 2005- 3 -(3)Find the pdf of minXi (4)Find the pdf of maxXi Ping Ma Lecture 19 Fall 2005- 4 -Ping Ma Lecture 19 Fall 2005- 5 -...
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