# hw7 - 36-226 Summer 2010 Homework 7 Due July 26 1. Now we...

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36-226 Summer 2010 Homework 7 Due July 26 1. Now we will prove that the CDF of any random variable is uniformly distributed on the interval [0 , 1]. We will do this in steps. Let X be a random variable with cdf F X ( x ) and inverse cdf F - 1 X ( x ). (a) Let Y = F X ( X ), a transformation of the random variable X . Show that Y U (0 , 1). Note: The assumption of invertibility is entirely unnecessary, however, it eases this particular method of proof. We can eliminate the assumption entirely and still get the result, but the proof is somewhat more diﬃcult. (b) Use the result that you have just proved to ﬁnd a pivotal quantity for X exp( θ ) and a 100(1 - α )% CI for θ . Notice that this is diﬀerent than both of the intervals in problem 4. 2. Let X 1 ,X 2 ,...,X n be a random sample from a Beta ( θ, 1) distribution. (a) Find an exact 100(1- α )% C.I. for θ . You may use the fact that Y = - ln X Exp(1 ). (b) Based on the following random sample from the above distribution, ﬁnd a 95% CI for θ using part (a) above. 0.48 0.56 0.36 0.68 0.70 0.76 0.64 0.25 (c) Using the same sample, ﬁnd a 95% CI for θ using the approximate CI for the MLE developed in class. (d) Compare the two intervals and explain why one is more appropriate than the other. 3. Let X 1 ,X 2 ,...,X n be a random sample from a Gamma (2 ) distribution. (a) Find an exact 100(1- α )% C.I. for θ . (b) Based on the following random sample from the above distribution and part (a), ﬁnd a 95% CI for θ 14 13 8 5 9 8 11 13 12 9 (c) Find an approximate 100(1- α )% CI for θ using the MLE.

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hw7 - 36-226 Summer 2010 Homework 7 Due July 26 1. Now we...

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