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Which is a one-to-one transformation from (x,y) to (r, 0). Note that 0 < r < +co and 0 < 6 < 2n. The Jacobian is given by: dx dx cosO -rs\nO dr 86 J = dy dy sin 9 rcosO ~dr dd In the inner integration of the sixth equality, again, integration by substitution is utilized, where transformation is s - —r1. Thus, we obtain the result 72 = 1 and accordingly we have 7 = 1 because of f(x) > 0. x~ / ^/2n is also taken as a probability density function. 1. Distribution Function: The distribution function (or the cumulative distribution function), denoted by F(x), is defined as: P(X

Let fa be the moment-generating function of /}. > where n and cr2 in (f>x(ff) is simply replaced by // 2"=1 a/ and cr2 £"=] a2 in 0^(0), respectively. Moreover, note as follows. , when ft = X is taken, ft = X is normally distributed as: X ~ N(JJ, a2In). 21 and this theorem. 1 Law of Large Numbers and Central Limit Theorem Chebyshev's Inequality In this section, we introduce Chebyshev's inequality, which enables us to find upper and lower bounds given a certain probability. 30 CHAPTER 1. , g(X) > 0.

4. 2. 2 (Section 1 . 1 . 1 ), all the possible values of X are 0, 1,2 and 3. (note that X denotes the number of heads when a die is cast three times). That is, x\ = 0, jc2 = 1, Jt3 = 2 and x$ = 3 are assigned in this case. Y = 1) and P(X = 2), note that each sample point is mutually exclusive. 5. 2. Continuous Random Variable and Probability Density Function: Whereas a discrete random variable assumes at most a countable set of possible values, a continuous random variable X takes any real number within an interval /.