Let X 1, X 2, ••• , X. be a random sample of size n from a normal distribution, • • X;-N(,u, a2), and define U = LX; and W = L Xf. i= 1 i= 1 (a) Find a statistic that is a function of U and Wand unbiased for the parameter e = 2tJ-5a2• (b) Find a statistic that is unbiased for a2 + f.12• (c) Let c be a constant, and define Y; = 1 if X;~ c and zero otherwise. Find a statistic that is a function of Y1, Y2, ••• , Y, and also unbiased for F x(c) = ( c : f.l).
- A new component is placed in service and nine spares are available. The times to failure in days are independent exponential variables, T; -EXP(100). 10 (a) What is the distribution of L T;? i= 1 (b) What is the probability that successful operation can be maintained for at least 1.5 years? Hint: Use Theorem 8.3.3 to transform to a chi-square variable. (c) How many spares would be needed to be 95% sure of successful operation for at least two years?
- Suppose that X -x2(m), Y -x2(n), and X and Y are independent. Is Y -X -X2 if n >m?
- Suppose that X-x2(m), S =X+ Y-x2(m + n), and X and Yare independent. Use MGFs to show that S -X -x2(n).
- A random sample of size n = 15 is drawn from EXP(O). Find c so that P[cX < 8] = 0.95, where X is the sample mean.
- The distance in feet by which a parachutist misses a target is D = Jxr +X~, where X 1 and X 2 are independent with X; -N(O, 25). Find P[D ~ 12.25 feet].
- Consider independent random variables Z;-N(O, 1), i = 1, … , 16, and let Z be the sample mean. Find: (a) P[Z < tJ. (b) P[Z1 -Z2 < 2]. (c) P[Z1 + Z2 < 2]. (d) PL~l Zf < 32 J. (e) PL~1 (Z;-Z)2 < 25 J.