This completes the proof of chebyshevs inequality for

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This completes the proof of Chebyshev’s inequality for the special case where the random variable X is continuous. A similar proof may be used in the discrete case. Along with Chebyshev’s inequality, we shall employ a few basic facts about the expected value and variance to prove the law of large numbers. You should remember these facts from your previous coursework. Let X and Y be arbitrary random variables, and let c be a constant. Then: 1. E ( cX ) = cE ( X ). 2. E ( X + Y ) = E ( X ) + E ( Y ). 3. Var( cX ) = c 2 Var( X ). 4. Var( X + Y ) = Var( X ) + Var( Y ) + 2Cov( X, Y ). Armed with Chebyshev’s inequality and our four basic facts, we may proceed to proving the law of large numbers. We need to show that lim n →∞ P ( | ¯ X n - μ | > ε ) = 0 for all ε > 0 . For any ε > 0, Chebyshev’s inequality tells us that P ( | ¯ X n - μ | > ε ) 1 ε 2 E ( ( ¯ X n - μ ) 2 ) . Therefore, to show that ¯ X n p μ , it is enough for us to show that lim n →∞ E ( ( ¯ X n - μ ) 2 ) = 0 . Using facts (1) and (2) above, we can see that E ( ¯ X n ) = E 1 n n X i =1 X i ! = 1 n E n X i =1 X i ! = 1 n n X i =1 E ( X i ) = 1 n n X i =1 μ = μ. Therefore, E ( ( ¯ X n - μ ) 2 ) = E ( ( ¯ X n - E ( ¯ X n )) 2 ) = Var( ¯ X n ) , 12
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and so to complete the proof, we need only show that lim n →∞ Var( ¯ X n ) = 0. Using fact (3) above, we can see that Var( ¯ X n ) = Var 1 n n X i =1 X i ! = 1 n 2 Var n X i =1 X i ! . Fact (4) tells us that the variance of the sum of the X i ’s is equal to the sum of the variances of the X i ’s, plus two times the sum of all the covariances between different X i ’s. But we assumed earlier that the X i ’s are iid, meaning in particular that all the X i ’s are independent of one another. Therefore, all of the covariances are equal to zero, and so the variance of the sum of the X i ’s is just the sum of the variances: Var n X i =1 X i ! = n X i =1 Var( X i ) = n X i =1 σ 2 = 2 . We have now established that Var( ¯ X n ) = 1 n 2 Var n X i =1 X i ! = σ 2 n , which converges to zero as n → ∞ . This completes our proof of the law of large numbers. 10 Convergence in distribution Suppose we have an infinite sequence of random variables Z 1 , Z 2 , Z 3 , . . . . Previously, we used the notion of convergence in probability to describe the possibility that this sequence of random variables may appear to be converging to some constant value. Convergence in distribution describes a different kind of convergence. Suppose that each Z n has cdf F n . If there is a random variable Z with cdf F , and if the cdfs F n become arbitrarily close to the cdf F as n → ∞ , then Z n is said to converge in distribution to Z . To be more precise, the sequence of random variables Z 1 , Z 2 , Z 3 , . . . is said to converge in distribution to Z if lim n →∞ F n ( x ) = F ( x ) for all x. We write this as Z n d Z . For a simple example of convergence in distribution, suppose that each Z n has the U (0 , 1 + 1 /n ) distribution. In this case, the cdf of Z n resembles the cdf of the U (0 , 1) distribution more and more closely as n → ∞ . The cdf of Z n is given by F n ( x ) = 0 for x < 0 nx n +1 for 0 x 1 + 1 n 1 for x > 1 + 1 n , 13
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while the cdf of the U (0 , 1) distribution is given by F ( x ) = 0 for x < 0 x for 0 x 1
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