STAT 233 Final, Spring 2005

L. 2. 3. AMERICAN UNIVERSITY OF BEIRUT MATH 233, Final Exam February 2, 2002 Time = 1 hour 30 minutes Front au ordinary deck of 52 playiug cards, cards are drawn oue at a time aud without replacement. Compute the probability that the third spade appears on the sixth draw.(5 pts) Bowl I contains 7 ved balls and 3 white balls and bowl IL has 4 red and 6 white balls. Two balls are selected randomly and without replacement from Bowl I and transferred to bowl II. Three balls are then selected at, random and without replacement from bow! I. (a) Find the probability of sclecting three white balls. (5 pts) (b) Given three white balls are selected from bowl I, compute the conditional probability that two white balls were transferred from bowl I. (5 pts) Let X and Y have the joint pdf f(x,y) = kay if 0 <2 <y <1, zero elsewhere. (a) What is the value of & such that f(z,y) is a pdf? (5 pts) (b) Find the value of the random variable E(X|y). (10 pts) Let X and Y he iucependent random variables with respective distri- sutions b(10,1/3) and b(15,1/3). If Z = 25 — X - Y, what is the wobability of Z exceeding 2? (10 pts) t is known from the past that the birth weight (in Kg) of male baby in Lebanon has a N(3.5,1) distribution. Let Y denote the number of yabies in a randoin sample of 20 new born male babies to exceed 3.5 kg.. Find P(Y < 3). (10 pts) 6. Let X and Y be independent and identically distributed random vari- ables with common distribution N(0,1). Let U = X2+ Y? and V = X+Y. (a) Find the joint pdf, g(u,v), of (U,V). (10 pts) (b) Find the marginal density function of U. (5 pts) (¢) Compute the covariance of U and V. (5 pts)