Question

A jar contains 6 white beads and 3 black beads. Beads are chosen randomly from the jar one at a time until the third time a black bead turns up. a. Suppose that each bead is replaced before the next is chosen. How many beads should you expect to be chosen in the course of the experiment? [5] b. Suppose that if a bead is white, it is not replaced before the next bead is chosen, but if it is black, it is replaced before the next is chosen. How many beads should you expect to be chosen in the course of the experiment? (10) t<0 Let g(t) = { 2te ſ 2te 120 be the probability density function of the continuous random variable X. a. Verify that g(t) is indeed a probability density function. (8) b. Find the median of X, i.e. the number m such that P(x 5 m)= { = 0.5. (7)

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Answer 1

given that far ? Solm: In problem we have Contains 6 white beads total g beads 3 black beads and Beads are chosen rondomly from the jour one at a time and define event that the third ble draw bead until the third time a black bead turn up. Chosen, (a) Here each bead is suplaced before the next is hele have to find how many beads should be expect to be cholen in Cowere ay experiment ez? So when we draw first draw is black exp end Lipet draw white Second draw block; upend I coow White, a draw white, 3 craw block; expand again f I draw white, a draw white, 3 draw, white, 4 draw black then the nud & ball Similarly į draw white, 2draw block, 3 draw while, a draw black, sdraw white, 6 draw white, 7 draw block so we need y Balls for maximise o New Condition - Ha bead is white it is not veplaced before the next draw but ij it is black it I's replaced before the next draw g beads Conduction the experiment = I kthite, 2 black, 3 White, 4 black Replace iv (b)r Replace 1 So no ར་ 2

) 5 White, 6 Black 7 white, & Block, Replace Replace ,g white, lop White, Il Block, exp- and So ble need total 11 Balls for this event. Now Again we have sto dit) = patetit? be pdf of Cont. rux lo sto @ for pony fun to pdf will need to satisfy following (v) git) to be nom Negative in; to (10) sa Altdt21 so Condition Ci) s fullfill it is the and ēt is non Megative dt= 2 So Ct) is non Negative in the (13T2të fötz -t at an " Yet t = 2 at alt zdz e o dz 2 z-(02)) = -(0-1) = ! So gle) is pdf.

(6) for Medion of x P(x.<m) =0.5 So eot saudtz 0.5 => meritētadt = "ste at = Zet ť: az at dt zdz Smeldz z 1 C9)? z emas a) I-em? let emek; taking Log - m'c (1930:5) =) + m +10:3010 mz 0.3010 m20.5486 ) A median of x.