Download Assignment Booklet Bachelor`s Degree Programme Probability and

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MTE -11
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Assignment Booklet
Bachelor’s Degree Programme
Probability and Statistics
School of Sciences
Indira Gandhi National Open University
New Delhi
2003
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Dear Student,
We hope you are familiar with the system of evaluation to be followed for the Bachelor’s Degree
Programme. At this stage you may probably like to re-read the section of assignments in the Programme
Guide for Elective Courses that we sent you after your enrolment. A weightage of 30 per cent, as you are
aware, has been earmarked for continuous evaluation, which would consist of two tutor-marked
assignments for this course. Both these assignments are in this booklet.
Instructions for Formating Your Assignments
Before attempting the assignment please read the following instructions carefully.
1)
On top of the first page of your answer sheet, please write the details exactly in the
following format:
ROLL NO : . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NAME : . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ADDRESS : . . . . . . . . . . . . . . . . . . . . . . . . . . . .
............................
............................
COURSE CODE : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
............................
COURSE TITLE : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ASSIGNMENT NO. : . . . . . . . . . . . . . . . . . . . . . . . . . . .
STUDY CENTRE : . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DATE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO
AVOID DELAY.
2)
Use only foolscap size writing paper (but not of very thin variety) for writing your answers.
3)
Leave a 4 cm. margin on the left, top and bottom of your answer sheet.
4)
Your answers should be precise.
5)
While solving problems, clearly indicate which part of which question is being solved.
6)
These assignments are to be submitted to the Study Centre as follows:
•
Assignment 1 should be submitted to the Study Centre Coordinator within 6 weeks of
receiving this booklet.
•
Assignment 2 should be submitted to the Study Centre Coordinator within 12 weeks of
receiving this booklet.
Answer sheets received after the due date shall not be accepted.
We strongly suggest that you retain a copy of your answer sheets.
Wish you good luck.
2
ASSIGNMENT-1
(To be done after studying Blocks 1 and 2.)
Course Code: MTE -11
Assignment Code: MTE-11/TMA-1/2003
Maximum Marks: 100
1)
The following table gives the monthly wages of 200 employees working in different companies of a
city.
Monthly wages (Rs. in ‘000)
Number of Employees
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
12
18
25
40
54
28
15
08
Calculate the average monthly wage of the employee of this group. Also find the limits for the
monthly wages (lower and upper) under which 50% of the employees fall.
(10)
2)
a)
The average sales (in units) and the variance (in units) of a product in three different stores in a
specific locality is given below.
Store
Average sales
Variance
Number of days
A
12
05
12
B
10
07
10
C
08
03
09
Calculate the average sales and variance of all the three groups combined together.
b)
6 observations on (x, y) yielded the following data:
P
P
xi = 30,
P
x2i = 200,
yi = 180,
P
Σxi yi = 1000
y2i = 5642
i)
Determine the correlation coefficient between x and y.
ii)
Given x = 10, what will be the predicted value of y.
iii) Given y = 15, predict x.
3)
a)
(5)
(5)
The following is the record about the number of defectives observed in the manufacturing process during 10 days period of a manufacturing firm.
12
19
13
18
14
10
29
15
11
17
Compute the first four moments about the arithmetic mean and the coefficient of kurtosis for the
data.
(7)
3
4)
b)
In a certain community, 10% of all people above 50 years of age have diabetes. A health service
in this community correctly diagnoses 95% of all persons with diabetes as having the disease,
and incorrectly diagnoses 5% of all persons without diabetes as having the disease. Find the
probability that a person randomly selected from among all people of age above 50 and diagonised by the health service as having diabetes actually has the disease.
(3)
a)
Calculate the Product moment correlation coefficient between the two variables sales (x) and the
advertising expenditure (y) for the following data.
Sales(x)
0 − 1000
1000 − 2000
3000 − 4000
Adv. Expenditure (y)
100-150 150-200 200-250
1
7
13
10
15
16
15
20
19
(8)
b)
Let X be a r.v. with p.m.f. given by the following table
−2
−1
0
1
2
3
20
4
20
6
20
4
20
3
20
Compute E(X2 )
5)
(2)
The following table shows the data on the monthly income and the expenditure on entertainment
(figures in 100 rupees) of 100 families in a specific locality.
Income
Expenditure
40
2
45
3
54
4
61
5
65
5
63
6.5
70
7
Fit a regression line which can be used to estimate the expenditure for the known income. Also
estimate the likely expenditure if a family has an income of Rs. 7500.
(10)
6)
a)
There are only two candidates A and B who are competing for a post of Company Executive
(CE) in a pharmaceutical company. The respective probabilities for A and B to be selected as
CE are 0.6 and 0.4. The company wants to introduce a new drug after recruiting the CE. From
the interview of both the candidates it has been observed that the chances of introducing the
drug by A is 45% and by B is 55%. Calculate the probability that the drug will be
introduced.
(5)
b)
The probability of having a male child is 0.4 in a family of 4 children. Find the probability that
i)
there are 2 sons
ii)
there are 4 daughters
If 100 families, each with 4 children are interviewed, find how many families would you expect
to have
c)
i)
2 sons
ii)
4 daughters
(3)
For a Poisson distributed random variable X, P[X = 4] = P[X = 5]. What is the mean and
variance of the distribution.
(2)
4
7)
a)
A box contains m white and n black balls the total number of balls being m + n. Suppose r balls
are drawn one by one (without being returned to the box), what is the dispersion of the number
of white balls drawn?
(10)
b)
An investor wishes to invest in equity shares of three companies A, B and C. He calculated that
his expected earnings, variance and co-variances are as follows:
Company
A
B
C
Expected
earnings(in%)
10
15
08
Variances
40
85
12
Co-variances
Cov(A, B) = 30
Cov(B, C) = −25
Cov(A, C) = 2
He invests 0.5 percent of his capital in the shares of company C, 0.3 percent of his capital in A
and 0.2 percent of his capital in B. Find his total expected earnings and variance of the
earnings.
(3)
8)
c)
A chemical plant holds for all its employees to decide whether or not to accept a new pay deal.
80% of the employees voted and it is known that 60% of the employees are union members. The
union ascertains that 90% of the employees are either union members or voted (or both). What
is the probability that an employee selected at random is a union member who voted?
(2)
a)
From records of 10 Russian army corps kept over 20 years, the following data was obtained
showing the number of deaths caused by the kicks of a horse. Determine the average number of
deaths per army corps per annum, and calculate the theoretical Poisson frequencies and compare
with the observed frequencies.
Number of deaths per
0
1
2
3 4
Army corps per annum
(8)
Frequency of occurrence 109 65 22 3 1
b)
The mean and the standard deviation of family income (in Rs.) in a year of 2 regions A and B
are as follows:
Region
A
B
Mean
320 370
Standard deviation
54
60
Which of these 2 regions shows a greater disparity in family income? Justify your answer. (2)
9)
a)
Reena enters into three independent ventures. The probability of success in these three ventures
are 0.5, 0.25 and 0.1 respectively. What is the probability that she will succeed in exactly three
ventures.
(5)
b)
5 books are selected at random from a shelf having 4 books of History, 7 books of Economics
and 10 books of English, what is the probability that
i)
no English book is selected?
ii)
one book of History, two books of Economics and two books of English will be
selected.
(5)
10) Which of the following statements are true? Give reasons for your answers.
i)
ii)
2
5
The abscissa of the point of intersection of the two Ogives of less-than type and more-than type
gives the mean of the data.
If P(A) = 0.5, P(A ∪ B) = 0.7 and A and B are independent events, then P(B) =
5
iii) The least-square line always passes through the point (x̄, ȳ).
iv) For two events A and B, it is known that P(A) = 0 and P(B) > 0, then P(A|B) > 0
v)
If x1 , x2 , . . . , xn are n observations of a variable x, then n
n
X
i=1
6
xi ≥
n
X
i=1
xi
!2
(10)
ASSIGNMENT-2
(To be done after studying Blocks 3, and 4.)
Course Code: MTE -11
Assignment Code: MTE-11/TMA-2/2003
Maximum Marks: 100
1)
a)
Suppose a random variable X has the following density function
f(x) =
(
ae−ax for x ≥ 0 and a > 0
0
otherwise
Find the moment generating function. Also obtain the mean and variance of the
distribution.
b)
(4)
Given that the joint density of two random variables X and Y is
f(x, y) =
2
,
(1 + x + y)3
x > 0, y > 0
Find
c)
i)
fX (x)
ii)
fY|X (y|x)
(3)
Show that for any two random variables X and Y with Var.(X) < ∞
Var.(X) = E[Var.(X|Y)] + Var[E(X|Y)]
2)
a)
(3)
A variable X has the density
f(x) = 24x−4 ,
x≥2
Compute the upper bound for P[|X − µ| > δ] using chebychev’s inequality where δ is a
constant.
b)
3)
a)
It is known that 40% of the population die within a year. 30 persons are vaccinated of which
only 5 die in a year. The inventor of the vaccine claims that the vaccine has reduced the mortality
by 20%
i)
Discuss whether the vaccine is effective.
ii)
n
Find the smallest value of n i.e. sample size such that if die after vaccination, his claim
6
is justified.
(5)
The dynamo bulbs manufactured for cycles have an average life of 400 hours with a standard
deviation of 250 hours. Out of 500 bulbs thus manufactured
i)
how many bulbs would have burnt more than 600 hours?
ii)
less than 300 hours?
(Assume normal distribution)
b)
(5)
(5)
Let X1 , X2 , X3 , . . . Xn denote random sample of size n from uniform distribution with probability density function


1
1
≤ x ≤ θ + and − ∞ < θ < +∞
f(x, θ) =
2
2
 0, elsewhere
1, θ −
7
Obtain Maximum Likelihood Estimator for θ.
4)
a)
(5)
Let p be the probability that a coin will fall head in single toss in order to test
H0 : P =
1
2
against
H1 : p =
3
4
The coin is tossed 5 times and H0 is rejected if more than 3 heads are obtained. Find the
probability of Type I error and the power of the test.
(5)
b)
Consider the function f(x) given by
f(x) =
x<0
0≤x<1

0,





 x,
1
, 1≤x<2


2




x≥2
0,
i)
Sketch f(x) and show that it satisfies the condition to be a probability density function of a
continuous random variable X.
ii)
Find the cumulative distribution function F(x)
iii) Calculate the mean and the variance of X.
5)
a)
(5)
Survey has been conducted to study the effectiveness of a specific advertising media (Television)
for a product. Data have been collected before and after the advertisement on number units sold
in a given locality from a sample of 10 customers as given below.
Before
After
:
:
10
12
12
11
5
7
4
6
15
14
9
7
7
12
10
15
13
14
10
11
Test the null hypothesis that the advertisement is effective i.e. H 0 : µ1 − µ2 = 0 against the
(6)
alternative, H1 : µ1 − µ2 > 0 at level α = 0.05.
b)
6)
Fifty measurements of the acceleration due to gravity g, had a mean value of 9.8 ms −2 and s.d.
of 0.75 ms−2 (assuming that the gravity g follows normal distribution)
i)
What is the 95% confidence interval of g?
ii)
How many measurements would be necessary to reduce the 95% confidence interval to
9.7 < µ < 9.9? [Given that Z0.025 = 1.96, Z0.05 = 1.647]
(4)
a)
A sample of 20 students taken from a larger group of students showed the variability in understanding a specific concept in mathematics as 12.73. Predict 90% confidence interval for the
variance of group consisting of all the students.
(4)
b)
Suppose (X, Y) has the joint probability density function
f(x, y) =
Find E(X|Y).

 x2 exp(−x(y + 1)),

0
x ≥ 0, y ≥ 0
otherwise
(6)
8
7)
a)
The joint probabiliy density function of X and Y is given by
f(x, y) =
(
k(1 − x − y), k a constant, when x + y ≤ 1, 1 ≥ y ≥ 0 and 1 ≥ x ≥ 0
0
elsewhere.
Find
i)
the marginal probability density function of X
ii)
the value of k.
iii) the probability density function of Y when X =
b)
1
2
A bus is due at a bus-stop at 11.10 a.m. Its actual time of arrival, Y is normally distributed with
a mean of 11.14 a.m. and standard deviation of 4 minutes. A regular passenger arrives at the
bus-stop at a time X, which is normally distributed with a mean of 11.09 a.m. and standard
deviation of 2 minutes.
i)
What is the probability that the passenger arrives after 11.10 a.m.?
ii)
What is the probability that the bus arrives before 11.10 a.m.?
iii) What is the probability that the passenger misses the bus?
8)
(6)
a)
If the population correlation
√ is zero (i.e. ρ = 0) the show that for the sample correla√ coefficient
tion coefficient r, t = r( n − 2)/( 1 − r2 ) is distributed as student t distribution with (n − 2)
degrees of freedom.
(8)
b)
Let (X, Y) have the following p.d.f
f(x, y) =
(
6xy2 , 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
0,
elsewhere
Find P(2Y + X ≤ 1)
9)
(4)
a)
(2)
Let X1 , X2 , X3 and X4 be independent random variables such that
E(Xi ) = µ and Var.(Xi ) = σ 2 for i = 1, 2, 3, 4.
Three different statistics are defined as
Y=
X1 + X 2 + X 3 + X 4
4
Z=
X1 + X 2 + X 3 + X 4
5
T=
X1 + 2X2 + X3 − X4
4
i)
Examine whether Y, Z and T are unbiased estimators of µ.
ii)
What is the efficiency of Y relative to Z?
iii) Find the correlation coefficient between Y and Z using the efficiency parameter.
9
(5)
b)
From a population of 540, a sample of 60 individuals is taken. From this sample, the mean is
found to 6.2 and standard deviation is 1.368.
i)
Find the estimated standard error of mean.
ii)
Construct 96% confidence interval for the population mean.
(5)
10) Which of the following statements are true and which are false? Give reasons for your answers.
i)
If X is a beta random variable with the parameters α and β, then 1 − X is a beta random variable
with parameters β and α.
ii)
If T = θu where θ is an unknown parameter and u is a random variable having a chi-square
T
distribution with 2 degrees of freedom, then is an unbiased estimator of θ.
2
iii) If X has the distribution N(10, 9) and if Y = 5X + 3, then Y has distribution N(53, 81).
iv) In a problem of testing of simple hypothesis against a simple alternative, if the probability of
type 1 error is known to be 0.06, then the power of the test will be 0.94.
v)
If γxy = 0.62, u = 5 + 6x and v = 7 − 3y, then γuv = 0.62
10
(10)