This document outlines the instructions and questions for a final exam in engineering. It allows basic calculating tools and states that students should show their work and justify their answers. Students should complete 6 of the 7 questions provided, with each question worth 10 marks. The questions cover topics in probability, statistics, and random variables including moment generating functions, joint and marginal distributions, confidence intervals, and the Poisson distribution.

1. F i n a l - E N G R 371 - A p r i l 1999
P e n s , p e n c i l s , erasers, and straight edges allowed. N o b o o k s . N o crib sheets. C a l c u -
lators allowed.
If y o u have a difficulty y o u m a y try making R E A S O N A B L E assumptions. State
t h e a s s u m p t i o n and h o w that a s s u m p t i o n limits y o u r answer. Show all your w o r k and
j u s t i f y all y o u r answers. Marks are given for h o w an answer is arrived at not j u s t t h e
answer itself.
D o A N Y 6 of the 7 questions given.
M a r k s : Six questions, 10 marks each. Total 60 m a r k s .
I n d i c a t e clearly w h i c h six q u e s t i o n s y o u want m a r k e d .
1. R a n d o m variable X is u n i f o r m l y distributed. Specifically:
0.2, x = 0,1,2,3,4
/(*) = 0. otherwise
(a) F i n d the m o m e n t generating function of X.
( b ) Using the m o m e n t generating f u n c t i o n find t h e first, second and third
m o m e n t s of X.
2. G i v e n the joint pdf:
( , j 2e~( +
x y
0 < y < o o , 0 < x < oo, y<x
n X
' V )
~ 0, otherwise
and
Z = X + Y
F i n d the p d f of Z. N o t e X and Y are N O T i n d e p e n d e n t .
3. T w o r a n d o m variables X and Y have m e a n s E[X] — 2 and E[Y] = 0, variances
a — 1 and a — 4 and correlation coefficient pxy = 0.6.
A n e w r a n d o m variable is defined:
Z = X + Y + 4
F i n d the m e a n and variance of Z.
4. A c o n t r o l l e d satellite is k n o w n t o have an error (distance f r o m target) that
is n o r m a l l y distributed with m e a n zero and standard deviation 4 feet. The
m a n u f a c t u r e r of the satellite defines " s u c c e s s " as a firing in w h i c h the satellite
c o m e s within 10 feet of t h e target. C o m p u t e the p r o b a b i l i t y that the satellite
fails.

2. T h e m e a n breaking strength of a fabric A was f o u n d t o b e 25.2 p o u n d s per
square inch ( p s i ) , based o n a sample of 35 s p e c i m e n s . T h e standard deviation
of the s a m p l e was 5.2 psi.
30 s p e c i m e n s of fabric B were also tested. T h e m e a n breaking strength was
f o u n d to b e 28.5 psi and t h e sample standard deviation was 5.9 psi.
A s s u m e that the p o p u l a t i o n s for b o t h fabrics are n o r m a l .
(a) F i n d a 9 9 % confidence interval for the p o p u l a t i o n m e a n JJLA and p o p u l a t i o n
variance a for fabric A .
( b ) F i n d a 9 5 % confidence interval for the difference JJLA — I^B-
T w o r a n d o m variables X and Y have a joint distribution f(x,y) with region of
s u p p o r t given in the following figure:
y
F i g u r e 1: R e g i o n of Support
Call t h e region of support 71. T h e distribution is
_ / K, (x y)e1Z
:
0, otherwise
(a) D e t e r m i n e K.
( b ) D e t e r m i n e P(X > Y).
( c ) D e t e r m i n e the m a r g i n a l distributions of X and Y.
( d ) D e t e r m i n e the c o n d i t i o n a l distribution f(xy) for any value of Y.

3. 7. T h e n u m b e r of m a c h i n e failures per day in a certain plant has a Poisson dis-
t r i b u t i o n w i t h p a r a m e t e r Xt = 3. Present m a i n t e n a n c e facilities can repair 3
m a c h i n e s per day. Failures in excess of three are repaired b y a c o n t r a c t o r .
(a) O n a given day w h a t is t h e probability of having m a c h i n e ( s ) repaired by
a contractor?
( b ) If t h e m a i n t e n a n c e facilities c o u l d repair four m a c h i n e s a day, what would
b e the p r o b a b i l i t y of having m a c h i n e ( s ) repaired b y a c o n t r a c t o r ?
( c ) W h a t is t h e e x p e c t e d n u m b e r of machines that fail each d a y ?
( d ) W h a t is t h e e x p e c t e d n u m b e r of machines that are repaired in the plant
each d a y ? ( H i n t : B e careful).
(e) W h a t is t h e e x p e c t e d n u m b e r of machines that are repaired b y the c o n -
tractor each d a y ?

4. Some Useful Equations
Pi A I I?) P { B 1 A ) P { A )
P [ A 1 B )
~ P(B)
fix I v) = f [ X
' y )
n 1 y )
f(y)
a = E[{X ~ „ f) x
a X Y = E[(X - ix ){Y x - ixy))
PXY =
<7x<JY
P(fj, -kcr<X</j, + k a ) > l - - ^
n
= -n(n + 1)
1=1
N
1 _ r.n+1
fro 1-r
CO %
e x
1'
In
a; -
U 1
- 1
f(x) = pq ~ x l
e~ (Xt) M x
f{x) =
xl
f( ) x
- ~7^ e 2
°2
V Z7T<7
CO
a —1 ^
x ~ e- dx— a;
a l x
= (a-1)1
M (t) x = E{e ] tx
1 N
N . .
2=1
4
TV
5 2
= ~ -^0'
A' - !

5. 1*1 a_
- < (X x - X) 2 + Z /21
a
pq
P ~ Z / l — < P <P + Z /
a 2 a 2 TV
(N - 1) <? 2
(jV - 1)5 2
— ~ T < ^ X <
A Q/2 XI-CK/2