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Unformatted text preview: (a) For what real numbers α does there exist a constant c ( α ) such that the function f α ( x ) = c ( α ) x α , x > 1 is a PDF? (b) Suppose α is such that f α is a PDF. If X is a random variable having this PDF, for what positive real numbers β does E ( X β ) exist? 3 4. [10 pts.] Calculate the PDF corresponding to the DF F ( x ) = , x ≤ ( x + x 2 ) / 2 , < x < 1 1 , x ≥ 1 5. [10 pts.] Calculate the DF corresponding to the PDF f ( x ) = cos( x ) 2 ,π 2 < x < π 2 Deﬁne the DF on the whole real line. 4 6. [20 pts.] Suppose the random variable X has the PDF (not the DF) given by f ( x ) = 2 x, < x < 1 . Find the median (not the mean) of the distribution of X . 5...
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 Spring '02
 Staff
 Probability theory, probability density function

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