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PBT602S - PROBABILITY THEORY 2 - 2ND OPP - JULY 2023 |
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f
nAmlBIA unlVERSITY
0 F SCIEn CE Ano TECHn
FACULTYOF HEALTH, NATURAL RESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS, STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSAM
LEVEL: 6
COURSE CODE: PBT602S
COURSE NAME: Probability Theory 2
SESSION: JULY 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
SUPPLEMENTARY/ SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
Dr D. B. GEMECHU
MODERATOR:
Prof R. KUMAR
INSTRUCTIONS
1. There are 5 questions, answer ALL the questions by showing all the necessary steps.
2. Write clearly and neatly.
3. Number the answers clearly.
4. Round your answers to at least four decimal places, if applicable.
PERMISSIBLE MATERIALS
1. Nonprogrammable scientific calculators with no covers.
THIS QUESTION PAPER CONSISTS OF 3 PAGES {Including this front page)
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Question 1 [14 marks]
1.1. Briefly explain the following:
1.1.1. Probability function
[3]
1.1.2. Measure on sigma algebra a(S)
[2]
1.1.3. Measure on a 5B(S) algebra
[2]
1.2. Show that if m and n are two measures on 5B(S),then m + n is measure on 5B(S),where
(m + n)(A) = m(A) + n(A)
[4]
1.3. Lets ={a,b,c},thenfind:
1.3.1. Power set, JJ(S)
[2]
1.3.2. Size of JJ(S)
[1]
Question 2 [24 marks]
2.1. Let X be a continuous random variable with p.d.f. given by
x+ 1, for - 1 < x < 0,
I f(x) = 1- x, for 0::; x < 1,
0, otherwise.
Then find cumulative density function of X.
[7]
2.2. A hole is drilled in a sheet-metal component, and then a shaft is inserted through the hole. The
shaft clearance is equal to the difference between the radius of the hole and the radius of the
shaft. Let the random variable X denote the clearance, in millimeters. The probability density
function of X is
fx(x) = {C(1 - x 4 ) for 0 < _x < 1
0
otherwise
2.2.1. Show that the value of C = 1.25.
[3]
2.2.2. What is the probability that the clearance is between 0.3 mm and 0.8 mm?
[4]
2.2.3. If R = X + 0.3, then find the expected value of R.
[3]
2.3. The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous
random variable X with cumulative distribution function (c.d.f.)
Fx(x) = {o1 - e-ax
for x < o
for x :2:0
Then use the c.d.f. of X to find
2.3.1. P(1 < X ::; 2)
[2]
2.3.2. the median value of X
[3]
2.3.3. the p.d.f. of X.
[2]
Question 3 [18 marks]
3.1. Suppose that the joint p.d.f. of two continuous random variables X and Y is given by
[; (x ) = {12x, 0 < y < x < 1; 0 < x 2 < y < 1,
XY 'y
0, elsewhere.
Find the marginal p.d.f. of Y.
[3]
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3.2. Let Y1 , Y2 , and Y3 be three continuous random variables with the following joint p.d.f.
= _ {6e-CY1+ 2Y2+3y3), for Yi > 0; (i l, 2, 3),
f(Yi,Y2,Y3) - 0,
e lsew here.
Then find
3.2.1. the marginal joint p.d.f of Y1 and Y3 . Hint: just find f (y 11 y 3 ).
(4]
3.2.2. the conditional distribution of Y2 given Y1 = 1, Y3 = l.
[3]
3.2.3. P(Y2 < 2IY1 = 1, Y3= 1).
[3]
3.3. If X and Y are linearly related, in the sense that Y = aX + b, where a > 0, then show that
PxY = l.
[5]
QUESTION 4 [28 marks)
4.1. Let a random variable Z follows a standard normal distribution [i.e., Z~N(0, 1)] with a p.d.f
given by
= f(z)
1
r-c.e-2
12
2
for
-
oo < z < oo
v2rr
4.1.1. Show that the moment generating function of Z is given by Mz(t)
=
1
ei
2
.
[8]
= 4.1.2. If X~N(µ,0' 2 ) and Z x-µ, then show that the moment generating function of Xis
(l
= 12 2
Mx(t) etµ+zt a . Hint: use the moment generating function of Z obtained above. [6]
4.1.3. Find the cumulant generating function of X and hence find the first cumulant.
[5]
4.2. Let X1,X 2 , .... Xn be independently distributed with normal distribution with mean µk and
= variance CY~,thus, Xk~N(µk, O'f). If Y If=1Xi,
. t2u2
= 4.2.1. Find the characteristics function of Y. Hint: ¢xk (t) eitµ--2-
[7]
4.2.2. Use the properties of characteristics function to comment on the distribution of Y. [2]
QUESTION 5 [26 marks)
= 5.1. Let Y be continuous random variable with a probability density function f (y) > 0. Also, let U
h(Y). If his increasing on the range of a given random variable, then show that
[6]
5.2. Let X1 and X2 be independent random variables with the joint probability density function given
by
if x1 > O; x 2 > 0,
otherwise.
= Find the joint probability density function of Y1 X1 + X2 and Y2
(10]
=== END OF PAPER===
TOTAL MARKS: 100
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