LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

        B.Sc. DEGREE EXAMINATION – MATHEMAT., PHYSICS & CHEMIST.

NO 15

FOURTH SEMESTER – APRIL 2008

ST 4206 / 4201 – MATHEMATICAL STATISTICS

 

 

 

Date : 28/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A                                     10×2 = 20 marks

                           Answer all questions

1. Define conditional probability
2. Define independent and mutually exclusive events
3. State the definition of random variable.
4. Let X have the pdf f(x) = 1/3, -1<x<2, find E(X)
5. State any two cases where poisson distribution can be applied.
6. Define the p.d.f. of continuous uniform distribution.
7. State any two applications of t-test
8. State Neyman – Pearson Lemma.
9. Define Maximum Likelihood Estimator.
10. Define Null and Alternative hypothesis.

                                      PART – B                                                5×8 = 40 marks

                              Answer any five questions

 

• State and prove Additional theorem of probability.
• An um contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that (i) two of the balls drawn are white (ii) one is of each colour (iii) none is red (iv) atleast one is white.
• Let X and Y be two r.v’s each taking three values – 1,0 & 1 and having the joint probability distribution

X Y	-1	0	1
-1	0	.1	.1
0	.2	.2	.2
1	0	.1	.1

(i) Show that X and Y have different expectations

(ii) Prove that X and Y are uncorrelated

(iii) Find Var (X) and Var (Y).

• From a bag containing 3 white and 5 black balls, 4 balls are transferred into an empty bag. From this bag a ball is drawn and is found to be white . what is the probability that out of four balls transferred ,3 are white and 1 is black ?
• Derive the MGF of Poisson distribution and hence obtain its mean and variance.
• Let the random variable X have the marginal density

f₁(x)=1,-1/2<x<1/2 and let the conditional density of Y given X=x be

f(y│x)= 1,   x<y<x+1, -1/2<x<0

=  1,   -x<y<1-x , 0<x<1/2 . Show that X and Y are uncorrelated.

• If X and Y are independent gamma variates with parameters µ and v respectively, show that the variables u=X+Y,Z=X/(X+Y) are independent and U is gamma variate with parameter (µ+ v ) and Z is a β₁ (µ, v ) variate
• Define the following (i) Unbiased ness (ii) Consistency (iii) Efficiency of an estimator

           PART – C                                     2×20 = 40 marks

                                  Answer any  two questions                                           

• a) State and prove Baye’s theorem

1. b) Derive the recurrence relation satisfied by the central moments of the

Poisson distribution.

• a) Suppose that two – dimensional continuous random variable (x,y) has joint d.f. given by f(x,y) = 6x² y, o<x<1, o<y<1,

=   0, otherwise

Find (i) P(0<X<3/4,1/3<Y<2),,(ii) P(X +Y <1)  (iii) P(X>Y)  (iv) P(X<1/Y<2)

1. b) State and prove Chebyshev’s inequality.

• a) Discuss the properties of normal distribution.(8)

1. b) The mean yield for one – acre plot is 662 kgs with s.d. of 32 kgs. Assuming normal diet, how many one – acre plots in a batch of 1000 plots would you expect to have yield, i) over700 kgs ii) below 150 kgs,

iii) what is the lowest yield of the best 100 plots?(12)

• a) Derive the probability density function of t-distribution with n degrees of freedom

1. b) In random sampling from a normal population N(µ, s²), find the estimators of the parameters by the method of moments.

 

 

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