Let X¯ be the sample mean of a random sample X1, . . . , Xn

| August 31, 2017

Question
Let X¯ be the sample mean of a random sample X1, . . . , Xn from the exponential distribution, Exp(?), with density function f(x) = (1/?) exp{?x/?}, x > 0.

(b) Using the mgf technique determine the distribution of X¯.

(c) Use (b) to show that Y = 2nX/? ¯ has a ?2 (chi-squared) distribution with 2n degrees of freedom.

(d) Based on (c), find a 95% confidence interval for ? if n = 10. (Hint: Find c and d such that P(c < [{2nX¯}/? ]<d ) = 0.95 and solve the inequalities for ?.)

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