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Unformatted text preview: gear? 10M Math351 Spring 2010 Exam 2 (1) (10 points) The cumulative distribution function F x of the random variable X
has the followmg graph: (3;) Find the following: P{X:0}= Va P{X:1}= O P{X=2}= 3/3— Va = VL
P{Xg2.5}: 3/;1 Em: O—L+gi%‘+3.}1~ :\.7§ (2) (15 points) The probability density function fx of a continuous random variable jCXm X has the following graph: 2 3 (a) Sketch the graph of FX, the cumulative distribution function of X em f (b) Calculate E[X 2 00

I. (x)0{x:jx —0(X+
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‘1 .1 Z L " Lt H ‘" L1 (3) (20 points) An urn contains 5 Red marbles, 3 Blue marbles and 2 Green marbles.
An experiment consists in taking a marble from the urn, recording its color and putting the marble back to the urn. The independent trials of this experiment are
performed. (a) What is the probability that in a series of 12 trials. Blue color will be recorded
exactly 4 times?1 3 9f— \ O W B e I (Lu(at Ga (b) What is the probability that the Blue color will be recorded for the ﬁrst time
at the 4”1 trial?
3 l s
10)'T‘ (c) What is the expected number of the trial when the Blue color is recorded
for the ﬁrst time? ‘ 0 ’ S/I'O 3 (d) What is the probability that the Blue color will be recorded for the third
time at the 10m trial? q_z ‘2.
9 . e a II ‘V ‘i. . q 3 l o 3 __ q
LAB—4'02?” (to) 5*(L
NM. 2354 (e) What is the expected number of the trial when the Blue color is recorded
for the third time? J0_
3—3—40 78 1111 Problems 3 and 4, just write a formula; for example, ( 43 ) ' 74 + 561 — 11 is an acceptable form of the answer, do not ﬁnish arithmetics. (4) (15 points) An urn contains 500 Red marbles 300 Blue marbles and 200 Green
marbles. An experiment consists in taking from the urn a sample of 10 marbles.2
(a) What is the expected number of Blue marbles in the sample? 300 1 3 $00 + 300 +200 {0' (b) What is the probability that the sample will contain exactly 2 Blue marbles?
300) 75 o)
1 ‘ 8
i o o 0
g m (c) What is the probability that the sample will contain 2 Blue marbles, 4 Red marbles and 4 Green marbles? 3 O o < 50 0)(2_06)
z ‘ ‘1 it
——'———‘———'—‘_'—_'_'—" woe
i0 (5) (15 points) An urn contains 1 Blue marble and 2 Green marbles and 997 Red
marbles. An experiment consists in taking a marble from the urn, recording its color
and putting the marble back to the urn. The independent trials of this experiment
are performed. Use Poisson approximation to estimate the probability of the event that in a
series of 100 trials, the Blue marble will appear at most two times. (Write a
formula involving exponents and not involving binomial coeﬂicients; use calculator
to arrive to the answer in decimal form. )
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N = 100 .— W W OE Jrﬁ‘ba
PFVHLXCY‘J‘M X ( havch in '
+r$ M WW m W >=N~P=OJ PiXéz} : P{x=oi+P{X:4§+P{x=2} g l )0 2One puts a hand into the urn and grabs 10 marbles at one time. 0.610% ( + 0,1 +0005) 2: Oﬂq‘is 2. '2.
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