exam1_100af10_solutions

# exam1_100af10_solutions - "r056 lG-l : .,_ ,-.‘...

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Unformatted text preview: "r056 lG-l : .,_ ,-.‘ University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Exam 1 21 October 2010 SetuT/ONY Name: Problem 1 (15 points) Three Cards are identical in form except that both sides of the ﬁrst one are colored green7 both sides of the second card are colored blue, and one side of the third one is colored green and the other side blue. The three cards are mixed up in a hat, and one card is randomly selected and placed down on the ground. If the upper side of the chosen card is colored green7 What is the probability that the other side is colored blue? {’[8G a 63 (f Pm) {9(8006) M V. 19(GQGG) + P(Go6&)+ F (0:155) NCJI’SQ'WQ Problem 2 (35 points) Answer the following questions: a. Suppose P(A) = 0.3, P(B) = 0.6, and PM n B): 0.2. Find P(A’|B). ,,— . F M8) 05.9 s. FUHS): 7: ' b. You roll two dice repeatedly until you observe the sum of 10 or the sum of 7. What is the probability that the sum of 10 is obtained before the sum of 7? a in Q?) ~_ i)('0 m mail} 2‘ F08) w v 6 m i _. ,M/ P. ._ y s. -U 4906?) POW“) c. The probability mass function of a random variable X is given by p(z) = 9%: with a: = 0, 1, 2, . . .. Find the constant c. :i t) :f m X! 2% Ce 4 f{x77’): :1, ‘93 Q? + Problem 3 (25 points) Answer the following questions: a. Show that the determination of negative binomial probabilities can be simpliﬁed by making use of the identity 7‘ P(X : z) = —P(Y = r) as where X follows the negative binomial distribution and Y N b(z, 1)). As a reminder, in the negative binomial distribution, X represents the number of trials needed until 1" successes occur (each trial has probability of success p). r .X ,‘r 3(~~Y Y, ,1 k/ r p“ Aw giriiﬂ'?) : 7'2 ﬁlﬁm’ gm m X“ r xd’ i - K‘r W) : (wit/In?) , Hkﬂl ._.. b. A and B alternate rolling a pair of dice, stopping either when A rolls the sum 9 or when B rolls the sum 6. Assuming that A rolls ﬁrst, ﬁnd the probability that the ﬁnal roll is made by A. r u A wN’S‘ . P(ﬁi MIME): P(aw (mil/\$7) T " 5 i ’L 1 "3L ‘1’ i 36 f I I? 3 J c. Suppose the number X of internet users that visit a particular website follow the Poisson distribution with parameter A = 3 per minute. Compute P(X > 2|X > 1). n x ‘> r ) ‘ ' X91 ?(>(71/Xw> : P(P(><>/) ;_.@/X£1) - 4,;;i : _ﬂ#,/w—~ ’ row) [email protected]' 0 —} axe"? W 2 ,3 n 7/— , i , i-i ‘P + Z" 5 ': a W "'3 {*3 / 4,5 o / . E Problem 4 (25 points) Answer the following questions: a. Customers arrive independently at a cashier. The probability that a customer pays with cash is 40%. Find the proba- bility that the 12th customer is the 8th that pays with cash. Pong NQéwWH/E Ki/VO/‘v/AL/ ? . '7 m- P()<:(Z> : g3?! )G‘UgOgéq‘“: >G.Qb0,69 v» 63.62/36} , b. Cards are selected with replacement until the ﬁrst ace is found. Given that the ﬁrst ace is found on or after the 5th trial, what is the probability that the ﬁrst ace is found after the 8th trial? i[><‘>3>7<>x§’>:1>(><>8/x>q) i.” "P {ygqv mei‘amwéi‘l ﬁﬁe‘i’WTV w, (W l” 3 / y {7/ c. Ten cards are selected with replacement. What is the probability that 4 of them are clubs? . f u 6 Sinra): :aiag, ...
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## This note was uploaded on 02/11/2012 for the course STATS 100A 262303210 taught by Professor Wu during the Fall '09 term at UCLA.

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exam1_100af10_solutions - "r056 lG-l : .,_ ,-.‘...

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