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Sma 2230: Probability And Statistics Ii Question Paper
Sma 2230: Probability And Statistics Ii
Course:Bachelor Of Science In Information Technology
Institution: Dedan Kimathi University Of Technology question papers
Exam Year:2013
DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
University Examinations 2013/2014
EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE IN
INFORMATION TECHNOLOGY
SMA 2230: Probability and Statistics II
DATE: 7TH December 2013 TIME: 8.30 AM { 10.30 AM
Instructions: Answer QUESTION ONE and any other TWO QUESTIONS.
QUESTION ONE (30 marks) (COMPULSORY)
(a) De ne the following terms.
(i) Continuous random variable . [1 mark]
(ii) Probability density function. [1 mark]
(b) Marketing estimates that a new instrument of soil sample analysis will be very
successful,moderately successful or not successful with a probability of 0.3,0.6 and
0.1 respectively. The yearly revenue associated with the each of the possible out-
come are estimated to be Ksh 10million , 5million and 1million respectively. Let
X be the yearly revenue of the product . Determine:-
(i) The probability mass function of X. [1 marks]
(ii) The expected revenue . [2 marks]
(iii) The standard deviation of the revenue. [3 marks]
(c) The probability density function of the length of hinges of a computer''s cabinet is
f(x) = 1:25 for 74:6 < X < 75:4 mm, zero elsewhere. Determine
(i) P(X < 74:8). [2marks]
(ii) P(X < 74:8; orX > 75:2). [3marks]
(ii) A batch of 1000 such hinges are bought. If the required speci cations are
74:7 < X < 75:3 mm where X is the length of a hinge, how many hinges do
not meet the speci cations. [3 marks]
(d) Messages arrive to a computer server at a rate of 10 per hour. Determine the
length of an interval such that the probability that no messages arrive during this
interval is 0.90. [4 marks]
(e) The lines to the customer care of a mobile phone operator are occupied 40 percent
of the time. Assuming that the events that the lines are occupied on successive calls
are independent and 10 calls are placed to the customer care , nd the probabilty
that:-
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(i) Exactly 3 calls go through. [2 marks]
(ii) At least one call goes through. [2 marks]
(f) The speed of le transfer from a server to a personal computer is normally
distributed with a mean of 60 Kilobytes per second and a variance of 16 (Kilobytes per second)2.
What is the probability that a le selected at a speed of:-
(i) Less than 70 Kilobytes per second. [2 marks]
(ii) Between 52 and 68 Kilobytes per second. [3 marks]
QUESTION TWO (20 marks)
(a) State the conditions under which it is permissible to use the normal distribution
as an approximations to the poisson distribution. [2 marks]
(b) The number of calls per hour received in a switchboard follows a poisson distri-
bution with a mean of 30. Approximate the probability that the switchboard
receives:-
(i) More than 33 calls. [4 marks]
(ii) Between 20 and 30 chips defective. [4 marks]
(c) The masses of packets of sugar are normally distributed. In a large consignment
of packets 5% have masses greater than 510g and 2% have masses greater than
515g. Estimate the number of packets in a lot of 1000 packets that would have a
mass of between 510g and 515g. [10 marks]
QUESTION THREE (20 marks)
A random variable X is said to have exponential distribution if its density is given by
f(x) = e??x > 0, x > 0 .
(i) Obtain the moment generating function. [5 marks]
(ii) Determine the mean of this distribution using the m.g.f. [7 marks]
(iii) The CPU of a personal computer has a lifetime that is exponentially distributed
with a mean life of 6 years. You have owned such a CPU for 3 years. What is the
probability it fails in the next 3 years. [4 marks]
(iv) The university has owned 10 such CPU for the last 3 years what is the probability
that at least one fails in the next 3 years . [4 marks]
QUESTION FOUR (20 marks)
(a) A random variable X has a hypergeometric distribution .
(i) Write down the probability mass function of X. [2 marks]
(ii) Obtain the mean X. [4 marks]
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(iii) A batch of parts contains 100 parts from a local supplier and 200 parts from a
foreign supplier . If four parts are selected randomly and without replacement,
what is the probability that two or more parts in the sample are from the
local supplier?. [5 marks]
(b) A discrete random variable X has p.m.f. f(x),for a x b. If each of the values
in the range of X is multiplied by a constant k show that the e ect is to multiply
the mean of X by k and the variance of X by k2. [4 marks]
(c) The are random variables X and Y such that X b(2; p) and X b(4; p). If
P(X 1) = 5
9 nd P(Y 1) . [4 marks]
QUESTION FIVE (20 marks)
(a). De ne the following terms as used in statistical hypotheses testing
(i) Null hypothesis. [1 marks]
(ii) Type I and Type II error. [2 marks]
(iii) Critical region. [1 marks]
(b) Two catalysts are being analyzed to determine how they a ect the mean yield of
a chemical process . Speci cally catalyst I is currently in use , but catalyst II is
acceptable. Since catalyst II is cheaper , it should be adopted provided it does
not a ect the process yield. A test is run in the pilot plant and the results are as
shown below. Is there any di erence between the mean yields? Test at = 0:05.
[10 marks]
catalyst I 91.50 94.18 92.18 95.39 91.79 89.07 94.72 89.21
catalyst II 89.19 90.95 90.46 93.21 91.79 97.04 91.07 92.75
(c) The cumulative distribution function of a random variable X is given by
F(x) =
8<
:
0 x < 0
2x ?? x2 0 < x 1
1 x 1
Find the mean and the upper quartile of X . [6 marks]
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