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sta 2006 0809_lecturenotes_3

# sta 2006 0809_lecturenotes_3 - Convergence Concepts Theorem...

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Unformatted text preview: Convergence Concepts Theorem : ( Chebychev’s inequality ) If X is a random variable with mean μ and variance σ 2 , then for every k > 0, P ( | X- μ | ≥ kσ ) ≤ 1 k 2 . Corollary : If = kσ , then P ( | X- μ | ≥ ) ≤ σ 2 2 . 26 Examples 1. If X is a random variable with mean 33 and variance 16, use Chebyshev’s inequality to find (a) A lower bond for P (23 < X < 43). (b) An upper bound for P ( | X- 33 | ≥ 14). 2. If Y is b ( n, . 25), give a lower bound for P ( | Y n- . 25 | < . 05) when n = 500. 27 Definition : A sequence of random variables, X 1 , X 2 , ..., converges in probability to a random variable X if, for every > 0, lim n →∞ P ( | X n- X | ≥ ) = 0 , or, equivalently, lim n →∞ P ( | X n- X | < ) = 1 . Theorem 1. X n P → X, Y n P → Y = > X n ± Y n P → X ± Y 2. X n P → a, Y n P → b = > X n Y n P → ab where a, b are constants 3. X n P → 1 = > X- 1 n P → 1 4. X n P → a, Y n P → b = > X n Y- 1 n P → a/b where a, b are constants and b 6 = 0 5. X n P → X, and Y a random variable = > X n Y P → XY 6. X n P → X, Y n P → Y = > X n Y n P → XY Theorem : Let X n P → X , and g be a continuous function defined on R . Then g ( X n ) P → g ( X ) as n...
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sta 2006 0809_lecturenotes_3 - Convergence Concepts Theorem...

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