2. A random variable X has a probability density function (pdf) given by 10

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2 A Random Variable X Has A Probability Density Function Pdf Given By 10 R 1 C 13 3 F X 0 Elsewhere For Some 1 (44.76 KiB) Viewed 30 times

2. A random variable X has a probability density function (pdf) given by 10<r <1, c 13 <3, f(x) = 0, elsewhere, for some real constant c. (a) Determine the value of the constant c and sketch the graph of f(r). In the following questions, replace c by its numerical value computed in (a). (b) Find the cumulative distribution function (cdf) F(x) of X, and sketch its graph. (c) Find P(X)). (d) Find Pl} < X < $). (e) What is P(X = 1)? (f) Find the moment-generating function (mg) MX(t) of X. Justify why it is well defined for any real finite value of t. (g) Find the 75th percentile of the distribution. Namely, find the value of 0.75 so that P(X 70.7) = F(70.7) = 0.75. (h) Find the conditional probability P(X >|X > ).