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SMA 2230 PROBABILITY AND STATISTICS II
UNIVERSITY EXAMINATION 2013/2014
FIRST SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE (MATHEMATICS AND COMPUTER SCIENCE)
SMA 2230: PROBABILITY AND STATISTICS II
th
Time: 9.00 – 11.00am
Date: Monday 10 February 2014
INSTRUCTIONS
ANSWER QUESTION ONE AND ANY OTHER TWO QUESTIONS
Question One (30mks)
cc) Distinguish between the following.
(i)
Discrete variables and continuous variables.
(2mks)
(ii)
Binomial distribution and passon distribution
(2mks)
(iii)
Type I Error and Type II Error
(2mks)
dd) The sample space of a random experiment is {a,b,c,d,e,f} and each outcome is equally
likely. A random variable is defined as follows:
Outcome
a
b
0
c
0
1.5
d
e
f
1.5
2
3
Determine i) The probability mass function of
(4mks)
ii) The mean of .
(3mks)
ee) A random variable X has the exponential distribution
( )={
i)
−
0
0ℎ
>0
Show that the moment generating function of
is
( ) = (1 −
)⁄
−1
(5mks)
ii)
Find the mean of .
(4mks)
∽ N(0,1) and Pr( < ) = 0.7, Pr( > ) = 0.45
ff)
Find Pr( <
< )
(4mks)
gg) A binomial distribution has mean 6 and standard deviation 2. Calculate , ,
.
(4mks)
Question Two (20mks)
2 +1
a) i) verify that
={
25
= 0,1,2,3,4
0
ℎ
is a probability mass function, and hence determine
ii) Pr(2 ≤
(4mks)
≤ 4)
(3mks)
iii) Pr( > −10)
(3mks)
b) The probability density function of the time to failure of an electronic component in a
− ⁄
1000
computer is ( ) = { 1000
0
>0
ℎ
Determine the probability that
i)
A component lasts more than 3000 hours.
(3mks)
ii)
A component fails in the interval from 1000 to 2000 hours
(4mks)
iii)
A component fails before 1000 hours
(3mks)
Question Three (20mks)
a) A geometric random variable x has probability distribution.
−1
i)
;
= 1,2, … …
( ) = { (1 − )
0
ℎ
Show that the moment generating function is given by
((
)=
1 − (1 − )
(5mks)
ii)
Use
( ) to find the mean and variance of .
b) i) A random variable
has a p.d.f
(5mks)
2
0
Determine the probability density function of
( )={
0< < 1
ℎ
= 3 + 6 using the cumulative method
technique.
iii)
(5mks)
Suppose that
parameter
=
1
+
1
and
2 are
two independent poisson random variables with
1
and 2 respectively. Find the probability density function of
2
using the moment generating function techniques.
(5mks)
Question Four (20mks)
a) Let
denote the number of bits received in Error in a digital communication channel, and
assume that
= 0.01. If 1000 bits are
is a Binomial random variable with
transmitted. Determine
i)
Pr(
= 1)
ii)
Pr(  2)
(2mks)
iii)
Mean and variance of
(2mks)
(2mks)
b) (i) Suppose a random variable
with moment generating function
( )=
3 +4
2
Determine the mean and variance of
(4mks)
(ii) The line width for a semi-conductor. Manufacturing is assumed to be normally
distributed with a mean
0.5 m and standard duration of 0.05 m
i)
What is the probability that a line width is greater than 0.62 m .
ii)
What is the probability that a line width is between 0.47
iii)
The line width of 90% of the sample is below what value?
and 0.63
(6mks)
c) Use Integration by parts to show that
Γ( ) = ( − 1)Γ( − 1)
(4mks)
Question Five (20mks)
a) Distinguish between Null hypothesis and Alternative hypothesis
(4mks)
b) A manufacturer is interested in the output voltage of a power supply used in a P.C.
output, voltage is assumed to be normally distributed with standard duration 0.25 volts
and manufacturer wishes to test.
0:
Using
i)
/
1:
≠5
= 8 units
The acceptance region is 4.85  ̅ 5.15
Find the value of ∝
ii)
=5
(8mks)
Find the power of the test for detecting a true mean output voltage of 5.17volts.
(8mks)