Answer: Let U have a uniform distribution on the interval

Chapter 4, Problem 4.132

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Let U have a uniform distribution on the interval [0, 1]. Then observed values having this distribution can be obtained from a computers random number generator. Let X (1/)ln(1 U). a. Show that X has an exponential distribution with parameter . [Hint: The cdf of X is F(x) P(X x); X x is equivalent to U ?] b. How would you use part (a) and a random number generator to obtain observed values from an exponential distribution with parameter 10?The right-hand side of this equation is a linear function ofX. If the distribution of X is concentrated in an interval overwhich g() is approximately linear [e.g., x is approximatelylinear in (1, 2)], then the equation yields approximationsto E(g(X)) and V(g(X)).a. Give expressions for these approximations. [Hint: Userules of expected value and variance for a linear functionaX b.]b. If the voltage v across a medium is fixed but current I israndom, then resistance will also be a random variablerelated to I by R v/I. If I 20 and I .5, calculateapproximations to R and R.