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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 4 Moment The r-th moment about the origin of X: E The r-th moment about the expectation of X: E Moment Generating Function Given a random variable X, the moment generating function is defined by and we have 0 1. For a random variable X Binomial (n, p) As the moment generating function of X is 1 . Find E(X) and Var(X) by using the formula. 2. Prove that if X has mgf and , then Y has the mgf = . 3. If X and Y are independent random variables with mgf’s and and Z = X + Y, then on the common interval where both mgf’s exist. Markov Inequality Let f (X) be a nonnegative function of the random variable X. If E[f (X)] exists, then, for every positive constant c, E . Pr Chebyshev Inequality Let X be a random variable and c be a positive constant, then | Pr | , where E and Var . 4. If f √ , √3 0 , Chebyshev’s inequality. √3 elsewhere , please find P | | and check the answer by using 5. Suppose that it is known the number of items produced in a factory during a week is a random variable with mean 50. If the variance of a week’s production is known to equal to 25, what is the lower bound of the probability that this week’s production will be between 40 and 60? ...
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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