Download 6 Questions on Probabilistic Operations Research with Solution - Exam | ISE 3414 and more Exams Operational Research in PDF only on Docsity! ISE 3414 - Probabilistic Operations Research: Midterm Exam
Fall 99: 12-1pm. section
Dr. Bish
September 27, 1999, 50 minutes
NAME
ID NO:
This examination is open books and open notes. You will have 50 minutes. Please
show all your work, and circle your answers. To obtain any credit you must
show how you obtained the answer even if you used a calculator to actu-
ally compute it.
HAVE FUN!
1. pmf - CDF (17 points) 4
. two die experiments (20 points) 5
. skiers insurance problem (30 points) ) 28
. pdf - CDF (18 points) 14
. ball problem (8 points) g
mB Ww oN
on
6. True-False questions (7 points} u
a
SUM: 100 points
READ FIRST all the problems. Maybe the easiest one is not the first one!
a 1.(17 points) Suppose X is a discrete random variable with values 1,2,3,4. If the
0 cumulative distribution function (CDF) of X is given by :
0, ifz<0,
0, if0<2<1,
E ifl<e<2,
F(x) =
3} fa <3,
B, if3<a<4,
1, ife>4
(a) Find the probability r .ss function (pmf) of X, that is, find P(X = i), for
i=0,1,2,3,4. (5 points)
P(X =0)= 0 Y (\)
253 P(X=D= CX_
P(X =3)= of
rem d= 1 OM
wPeasxs3a= COX \ (4 points)
Pie one eed) Soret -
ble Oeste 0 (C)P(Q<X<3)= CX Mw (4 points)
e 4g the 4 (@Pa<X<))= Of (4 points)
4.{18 points) Suppose X is a random variable with the following probability density
%
function (pdf).
p
cei—z7), if <a2<l,
swe ={" Os
0, 7" otherwise
Lh : (a) What value of ¢ makes this a valid pdf? (6 points)
S
) is the cumulative distribution function (CDF) of X? (6 points)
OM
Fon: Pix #am)= Cc. o-0937(1- a) abe
FON) = coe Tyg PO]
) What is P(X > 1/2}? (3 points)
wc O
i-PeE AS te Gots Lh 049571 Oe pleore
% .
(c-ii) What is P(X > 1/2)? ° points) Save . £2e fey
Pez ays 1- PGE. ie Commle- 4] oarz: ‘
% /3 irs. (8 points) A ball is drawn from an urn containing three white and five black balls.
‘After the ball is drawn, it is then replaced and another ball is drawn. This goes on
indefinitely. What is the probability that of the first four balls drawn, exactly two
z
are white? Ne3B 0 BAT
olkees: (AY Y
2 6 CO10n) C.340% good
POr2>: O3Z0" LS
= E277 KY
See
6. (7 points) State whether each of the following statements is True or False (no
explanations are necessary).
One For any two events A and B, P(A| B) = P(B| A).
fii) ) If events A and B are disjoint events, then events A and B are also dependent
(not independent) events.
Frag (iii) Tf events A and B are independent events, then events A and B are also disjoint
events.
Fansst) If events A and B are independent, then P(AU B) = P(A) + P(B).
Frise (v) If Ac B, then P(A | B) =1.
@ ole Tue P(A| B)+P(A|B)=1. he
6) Tue Fase wil P(A| B) + P(A*| B) =1. -
ONY
° ao
i
I
\
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