Lecture4 - PROBABILITY DISTRIBUTIONS: (continued) The...

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PROBABILITY DISTRIBUTIONS: (continued) The Standard Normal Distribution. Consider the function g ( z )= e z 2 / 2 , where −∞ <z< . There is g ( z ) > 0 for all z and also Z e z 2 / 2 dz = 1 2 π . It follows that f ( z 1 2 π e z 2 / 2 constitutes a p.d.f., which we call the standard normal and which we may denote by N ( z ;0 , 1). The normal distribution, in general, is denoted by N ( x ; µ, σ 2 ); so, in this case, there are µ = 0 and σ 2 =1. The standard normal distribution is tabulated in the back of virtually every statistics textbook. Using the tables, we may establish conFdence intervals on the ranges of standard normal variates. ±or example, we can assert that we are 95 percent conFdent that z will fall in the interval [ 1 . 96 , 1 . 96]. There are inFnitely many 95% conFdence intervals that we could provide, but this one is the smallest. Only the standard Normal distribution is tabulated, because a non-standard Normal variate can be standardised; and hence the conFdence interval for all such variates can be obtained from the N (0 , 1) tables. The change of variables technique. Let x be a random variable with a known p.d.f. f ( x ) and let y = y ( x ) be a monotonic transformation of x such that the inverse function x = x ( y ) exists. Then, if A is an event deFned in terms of x , there is an equivalent event B deFned in terms of y such that if x A , then y = y ( x ) B and vice versa. Then, P ( A P ( B ) and, under very general conditions, we can Fnd the the p.d.f of y denoted by g ( y ). ±irst, consider the discrete random variable x f ( x ). Then, if y g ( y ), it must be the case that X y B g ( y X x A f ( x ) . But we may express x as a function of y , denoted by x = x ( y ). Therefore, X y B g ( y X y B f { x ( y ) } , whence g ( y f { x ( y ) } .
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Lecture4 - PROBABILITY DISTRIBUTIONS: (continued) The...

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