2004-05 PROBABILITY, STATISTICS & QUEUEING THEORY
First Question is Compulsory
Answer any four from the remaining
Answer all parts of any Question at one place.
Time: 3 Hrs.
Max. Marks: 100
a) State the axioms of probability.
b) Explain confident intervals in estimation.
c) Explain the method of least squares.
d) Explain Principle of least square.
e) Explain Type I and II errors.
2. a) State and prove Baye’s formula on conditional probability.
b) We are given three urns as follows:
Urn A contains 3 red and 5 white marbles
Urn B contains 2 red and 1 white marble
Urn C contains 2 red and 2 white marbles.
An urn is selected at random and a marble is drawn from the urn. If the Marble is red, what is the probability that it came from urn A?
3. a) Define mathematical expectation of a random variable. Show that the expectations of the sum of two random variables is equal to the sum of their expectations.
b) Suppose that a pair of dice are tossed and let the random variable X denote the sum of the points. Find the expectation of X.
4. a) Define the mean to failure of a component. For aq series systems show that 0 ≤ E(X) ≤ min [E(Xc)].
b) Derive Markov inequality. Hence or otherwise state and prove Chebychev inequality.
5. a) Find the moment generating function about origin of the normal distribution.
b) Prove that a linear combination of normal variate is also a normal variate.
6. a) Derive normal equations to fit y = a + bx by the method of least squares.
b) Fit a least squares parabola having the form y = a + bx + cx-2 to the following data:
X: 1.2 1.8 3.1 4.9 5.7 7.1 8.6 9.8
Y: 4.5 5.9 7.0 7.8 7.2 6.8 4.5 2.7
7. a) Show that the correlation coefficient lies between x and y -1 and +1
b) Calculate the correlation coefficient between x and y for the following data.
X: 65 66 67 67 68 69 70 72
Y: 67 68 65 68 72 72 69 71
8. Arrivals at a telephone booth are considered to be Poisson with an average time of 12 min. between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with mean 4 min.
a) Find the average number of persons waiting in the system.
b) What is the probability that a person arriving at the booth will have to wait in the queue?
c) What is the probability that it will take him more than 10 mm. altogether to wait for the phone and complete his call?
d) Estimates the fraction of the day when the phone will be in use.
e) The telephone department will install a second booth, when convinced that an arrival has to wait on the average for at least 3 min. for phone. By how much the flow of arrivals should increase in order to justify a second booth?
Note: please note that the following question papers from 2001 to 2004 old question paper model. The question paper modal has been changed form 2004.
PROBABILITY AND STATISTICS 2001
(Effective from the Admitted Batch of 2000-2001)
Time: Three hours
Maximum: 75 marks
Answer Question No. 1 and any other FOUR.
Answer each question at one place.
All questions carry equal marks.
1. (a) If A and B are any events show that p (A u B) = p(A) + p(B) - p (A B).
(b) Write notes on correlation.
(c) State Baye's formula for conditional probability.
(d) Distinguish between large and small samples.
(e) Explain different transform methods and their utility.
2. (a) Define probability generating function. Derive the probability generating function of a geometric distribution.
(b) The joint density function of two continuous random variables X and Y is
f(X, Y) = {c x y, 0 < x < 4, 1 < y < 5 } {0 otherwise}
3.a. Explain Random variable, its expectation and variance for discrete case.
b. If f(x) = {1/2(x+1), -1 < x < 1}, {0 elsewhere} represents the density of a random variable X, find E(X) and Var (X).
4. (a) Define the mean time to failure of a component. For a series system show that
0 ≤ E (X) ≤ min [IF2 (XI )]
(b) Derive the Markov inequality. Hence or otherwise state and prove Chebychev inequality.
5. (a) Explain the chief characteristics of normal distribution and normal probability curve.
(b) Find the mean deviation from the mean for normal distribution.
6. (a) Explain the following:
i. Errors of first and second kind
ii. The best critical region
iii. Level of significance
iv. Simple and composite hypothesis.
(b) Suppose that n observations X1, X2 ... Xn are made from a Poisson distribution with unknown parameter X, find the maximum likelihood estimate of X.
7. (a) Derive the normal equations for fitting an equation of the form y = ax2 +bx +c.
(b) Fit a least square line of the form y = a + bx to the following data:
X 3 5 6 8 9 11
Y 2 3 4 6 5 8
8. (a) Show that the correlation coefficient is independent of origin and scale.
(b) A computer while calculating correlation coefficient between two variables X and Y from 25 pairs of observations obtained the following results
n = 25, σX = 125, σX2 = 650, σY = 100, σY2 = 460, σXY = 508.
It was however later discovered at the time of checking that he had copied down the pairs
X Y
8 12
6 18
Obtain the correct 6 8 value of correlation coefficient.
First Question is Compulsory
Answer any four from the remaining
Answer all parts of any Question at one place.
Time: 3 Hrs.
Max. Marks: 100
a) State the axioms of probability.
b) Explain confident intervals in estimation.
c) Explain the method of least squares.
d) Explain Principle of least square.
e) Explain Type I and II errors.
2. a) State and prove Baye’s formula on conditional probability.
b) We are given three urns as follows:
Urn A contains 3 red and 5 white marbles
Urn B contains 2 red and 1 white marble
Urn C contains 2 red and 2 white marbles.
An urn is selected at random and a marble is drawn from the urn. If the Marble is red, what is the probability that it came from urn A?
3. a) Define mathematical expectation of a random variable. Show that the expectations of the sum of two random variables is equal to the sum of their expectations.
b) Suppose that a pair of dice are tossed and let the random variable X denote the sum of the points. Find the expectation of X.
4. a) Define the mean to failure of a component. For aq series systems show that 0 ≤ E(X) ≤ min [E(Xc)].
b) Derive Markov inequality. Hence or otherwise state and prove Chebychev inequality.
5. a) Find the moment generating function about origin of the normal distribution.
b) Prove that a linear combination of normal variate is also a normal variate.
6. a) Derive normal equations to fit y = a + bx by the method of least squares.
b) Fit a least squares parabola having the form y = a + bx + cx-2 to the following data:
X: 1.2 1.8 3.1 4.9 5.7 7.1 8.6 9.8
Y: 4.5 5.9 7.0 7.8 7.2 6.8 4.5 2.7
7. a) Show that the correlation coefficient lies between x and y -1 and +1
b) Calculate the correlation coefficient between x and y for the following data.
X: 65 66 67 67 68 69 70 72
Y: 67 68 65 68 72 72 69 71
8. Arrivals at a telephone booth are considered to be Poisson with an average time of 12 min. between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with mean 4 min.
a) Find the average number of persons waiting in the system.
b) What is the probability that a person arriving at the booth will have to wait in the queue?
c) What is the probability that it will take him more than 10 mm. altogether to wait for the phone and complete his call?
d) Estimates the fraction of the day when the phone will be in use.
e) The telephone department will install a second booth, when convinced that an arrival has to wait on the average for at least 3 min. for phone. By how much the flow of arrivals should increase in order to justify a second booth?
Note: please note that the following question papers from 2001 to 2004 old question paper model. The question paper modal has been changed form 2004.
PROBABILITY AND STATISTICS 2001
(Effective from the Admitted Batch of 2000-2001)
Time: Three hours
Maximum: 75 marks
Answer Question No. 1 and any other FOUR.
Answer each question at one place.
All questions carry equal marks.
1. (a) If A and B are any events show that p (A u B) = p(A) + p(B) - p (A B).
(b) Write notes on correlation.
(c) State Baye's formula for conditional probability.
(d) Distinguish between large and small samples.
(e) Explain different transform methods and their utility.
2. (a) Define probability generating function. Derive the probability generating function of a geometric distribution.
(b) The joint density function of two continuous random variables X and Y is
f(X, Y) = {c x y, 0 < x < 4, 1 < y < 5 } {0 otherwise}
3.a. Explain Random variable, its expectation and variance for discrete case.
b. If f(x) = {1/2(x+1), -1 < x < 1}, {0 elsewhere} represents the density of a random variable X, find E(X) and Var (X).
4. (a) Define the mean time to failure of a component. For a series system show that
0 ≤ E (X) ≤ min [IF2 (XI )]
(b) Derive the Markov inequality. Hence or otherwise state and prove Chebychev inequality.
5. (a) Explain the chief characteristics of normal distribution and normal probability curve.
(b) Find the mean deviation from the mean for normal distribution.
6. (a) Explain the following:
i. Errors of first and second kind
ii. The best critical region
iii. Level of significance
iv. Simple and composite hypothesis.
(b) Suppose that n observations X1, X2 ... Xn are made from a Poisson distribution with unknown parameter X, find the maximum likelihood estimate of X.
7. (a) Derive the normal equations for fitting an equation of the form y = ax2 +bx +c.
(b) Fit a least square line of the form y = a + bx to the following data:
X 3 5 6 8 9 11
Y 2 3 4 6 5 8
8. (a) Show that the correlation coefficient is independent of origin and scale.
(b) A computer while calculating correlation coefficient between two variables X and Y from 25 pairs of observations obtained the following results
n = 25, σX = 125, σX2 = 650, σY = 100, σY2 = 460, σXY = 508.
It was however later discovered at the time of checking that he had copied down the pairs
X Y
8 12
6 18
Obtain the correct 6 8 value of correlation coefficient.
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