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# final_old - NTHU MATH 2810 Final Examination Jan 8 2008...

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NTHU MATH 2810 Final Examination Jan 8, 2008 Note. There are 8 problems in total. The total score is 100pts. To ensure consideration for partial scores, write down intermediate steps where necessary. 1. ( 16pts, 2pts for each ) For the following statements, please answer true or false. If false, please explain why. (a) Let X be a continuous random variable, then P ( X A ) = 0 for any countable set A . (b) For a continuous random variable, the values of its probability density function (pdf) must be between 0 and 1. (c) Transformation by using Jacobian can be applied to ﬁnd the joint pdf when the mapping between two groups of n random variables is not one-to-one. (d) A random variable X with possible values 0 and 1 will have E ( X k ) = E ( X ) for k = 2 , 3 , 4 ,... . (e) Let X 1 ,...,X n be i.i.d. from a distribution with ﬁnite variance. The variance of X n = ( X 1 + ··· + X n ) /n always tends to zero as the sample size n increases to inﬁnity. (f) The correlation coeﬃcient of two independent random variables is zero. (g) If X and Y are uncorrelated, then E ( X | Y ) = E ( X ). (h) If X and Y are independent, then E ( XY ) = E ( X ) E ( Y ) and E ( X/Y ) = E ( X ) /E ( Y ). 2. ( 15pts, 3pts for each ) For each of the random variables X below, determine the type of distribution (i.e., Normal, Exponential, Gamma, Beta, Uniform, Poisson, Hypergeomet-

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final_old - NTHU MATH 2810 Final Examination Jan 8 2008...

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