MCA - 1105
2354-I/01
M.C.A. DEGREE EXAMINATION - 2001
First Year - First Semester
PROBABILITY AND STATISTICS
(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 (X
I )]
(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 X
1, X
2 ... X
n 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 = ax
2 +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, σX
2 = 650, σY = 100,
σY
2 = 460, σXY = 508.
It was however later discovered at the time of checking that he had copied down the pairs
Obtain the correct 6 8 value of correlation coefficient.
