MATH 230 Final, Fall 2001

AMERICAN UNIVERSITY OF BEIRUT MATHEMATICS 230 FINAL EXAMINATION Date: January 31, 2002 Time = 1 hour 30 minutes Please, print your naiue and ID nuwnber on both the question sheet and the booklet. You are allowed to use one formula sheet and a calculator. 1. An urn contains 10 red and 9 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the sixth ball drawn if the sampliug is clone (a) with replacement. [5 pts] (b) without replacement. [5 pts] wo . Bowl A contains two red chips, Bowl B contains two white chips, and Bowl C contains one red chip and one white chip. A bowl is selected at random (with vqual probabilities), then one chip is drawn from that bowl. (a) Compute the probability of selecting a white chip. [5 pts] (b) If the chip sclected is white, compute the conditional probability that the other chip in the bowl is red. [5 pts] 3. Let the pdf of the random variable X be f(r) = &(2°3)7 if x = 2,3... and zero elsewhere. (a) Find the value of & such that f(z) is a pdf? (4 prs! (b) Find the mean and variance of X. [6 pts] 4. Let X have the pdf fle) =o /( tev)? if -x <r < oc. Find the pf of the trauslormation Y = |X]. [10 pts] 5. Ler XY and ¥ be independent random variables with respective binomial distributions bt). p) and b(2.p). Find the distriburie: of the random variable B=, + ry — VY — 17. {10 pts] 6. Let X equal the birth weight (in Kg) of babies in Singapore. Assume that the distribution of X is N(3.1,1). Let ¥° equal the number of babies that weigh less than 3.1 kg at birth among 20 of these babies selected indepeudently. Find P(Y < 3). {10 prs’ 9