STAT 230 Final, Spring 1999

> \/Math 233 ‘Final Exam 1. (13 pts.) Consider a sequence of n nonoverlapping finitely small intervals. Changes that occur in the 7 intervals are independent and have the same probability of occurrence given by, where X = # occurrences in an interval, O < p,g,r <1landp+q+r=1; P(X =1)=p P(X =2)=¢q P(X = 3)=r. Find the following probabilities. a). P(total occurrences in the n intervals = n). b). P(total occurrences in the n intervals = n + 1). c). P(total occurrences in the n intervals = n + 2). d). P(total occurrences in the n intervals > n). 2. (10 pts.) Consider the following p.d.f. ofar.v. X. z/2 O0<2x<1 f(x) = 1/2 2-1 What is the d.f., F(z) = P(X < x), of X? 2/6 l<2r<2 _ “— 0 elsewhere 3. (10 pts.) Let the r.v. X have the following d.f. 0 z<0 F(z) = a O<2<2 What is the p.d.f. of X, f(x)? t-e2" 2>2. y 4. (7 pts.) Let X be ar.v. such that F(X") =r!, r=0,1,2,... a). Find the m.g.f. of X. b). What is the distribution of X? 1 5. (8 pts.) Let Y be ar.v. such that E(Y") = mr: 7=9,1,2,... a). Find the m.g.f. of Y. b). What is the distribution of Y? 6. (20 pts.) Let X be ar.v. with the following p.d.f. 3,2 — J et ~2<4<2 f(z) { 0 elsewhere. a). What is the p.d.f. of the r.v. Y = X?? b), What is the p.d.f. of the rv. V = |X|? 7. (20 pts.) Let X have a Poisson distribution with parameter An. Provide the details of the proof to show that the Central Limit Theorem is true, that is, show that W = st has a limiting distribution which is N(0,1) as n — oo. 8. (12 pts.) Let Xj,..., X, be ars. froma N(, 07) distribution and let cy, ..., Cr be real numbers. Identify the distributions of the following r.v.'s (it is not necessary to provide the details). a). Soo X; b). X iz c). yo ser d). 3 (Ki XY i=]