LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FourTH SEMESTER – APRIL 2003

ST 4201 / sTA 201  –  MATHEMATICAL STATISTICS

 

28.04.2003

9.00 – 12.00                                                                                                     Max : 100 Marks

                                                                PART – A                                       (10´ 2=20 marks)

      Answer ALL the questions.

 

1. Two dice are thrown. What is the probability that the sum of the numbers on the two dice is eight?
2. The probability that a customer will get a plumbing contract is and the probability that he will get an electric contract is 4/9. If the probability of getting at least one is 4/5,determine the probability that he will get both.
3. Consider 2 events A and B such that and . Verify whether the given statement is true (or) false. .
4. Define i)  independent events and ii)  mutually exclusive events.
5. State any four properties of a distribution function.
6. The random variable X has the following probability function

X = x	0	1	2	3	4	5	6	7
P (X=x)	0	k	2k	2k	3k	k2	2k2	7k2+k

Find k.

 

7. Let f (x) =

0    ;   else where

Find E(X).

8. Let X ~ B (2, p) and Y~B (4, p). If P , find P.
9. Define consistent estimator.
10. State Neyman – Pearson lemma.

 

 

                                                                PART – B                                         (5´ 8=40 marks)

      Answer any FIVE questions.

 

11. A candidate is selected for three posts. For the first post three are three candidates, for the

second there are 4 and for the third there are 2. What are the chances of his getting

1. i) at least one post and  ii)  exactly one post?
2. Three boxes contain 1 white, 2 red, 3 green ; 2 white, 3 red, 1 green and 3 white, 1 red, 2 green balls. A box is chosen at random and from it 2 balls are drawn at random. The balls so drawn happen to be white and red. What is the probability that they have come from the second box?
3. Find the conditional probability of getting five heads given that there are at least four heads, if a fair coin is tossed at random five independent times.
4. Derive the mean and variance of hyper-geometric distribution.
5. Let X be a random variable having the p.d.f

 

f(x) =

Find the m.g.f. of X and hence obtain the mean and variance of X.

16. If X is B(n,p), show that E= p and E.
17. Let X be  N(m,s²).  i)  Find b so that
18. ii) If P (X < 89) =0.90 and P(X < 94) =0.95, find m and s².
19. If X and Y are independent gamma variates with parameters m and n respectively,

Show that  ~ .

 

 

 

                                                               PART – C                                         (2´20=40 marks)

Answer any TWO questions

19. If the random variables x₁ and x₂ have the joint  p.d.f

f  (x_{1 ,}x₂) =

i ) find the conditional mean of X₁ given  X₂ and  ii)  the  correlation coefficient

between  X₁ and X_2.

20. a)  Find all the odd and even order  moments of Normal distribution.
21. Let (X,Y) have a bivariate normal distribution. Show that each marginal distribution

in normal.

21. a) Derive the p.d.f of F- variate with (n₁,n₂) d.f.
22. Find the g.f of exponential distribution.

• a) Let X₁, X₂, …. X_n  be a  random sample of size n from N (q,1) . Show that the sample

mean is an unbiased estimator of the parameter q.

1. Write a short note on:
2. i) null hypothesis ii) type I and type II errors iii)    standard error
3. iv) one -sided and two -sided tests.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI– 600 034

B.Sc. DEGREE EXAMINATION  –  MATHEMATICS

Fourth  SEMESTER  – NOVEMBER 2003

ST 4201/STA 201 MATHEMATICAL  STATISTICS

14.11.2003                                                                                        Max: 100 Marks

9.00 – 12.00

 

SECTION – A                                             

Answer ALL the questions.                                                      (10 ´ 2 = 20 Marks)

1. Define an event and probability of an event.
2. If A and B any two events, show that P (AÇB^C) = P(A) – P(AÇB).
3. State Baye’s theorem.
4. Define Random variable and p.d.f of a random variable.
5. State the properties of distribution function.
7. Define marginal and conditional p.d.fs.
8. Examine the validity of the given Statement “X is a Binomial variate with

mean 10 and S.D  4”.

9. Find the d.f of exponential distribution.
10. Define consistent estimator.

 

SECTION – B                          (5 ´ 8 = 40 Marks)

Answer any FIVE questions.

 

11. An urn contains 6 red, 4 white and 5 black balls.  4 balls are drawn at random.

Find the probability that the sample contains at least one ball of each colour.

12. Three persons A,B and C are simultaneously shooting. Probability of A hit the

target is  ;  that for B is    and for C is  Find   i)  the probability that

exactly one of them will hit the target ii) the probability that at least one of them

will hit the target.

13. Let the random variable X have the p.d.f

 

 

Find P( ½ < X <  ¾) and    ii) P ( – ½ < X< ½).

14. Find the median and mode of the distribution

.

 

 

 

 

 

15. Find the m.g.f of Poisson distribution and hence obtain its mean and variance.

 

16. If X and Y are two independent Gamma variates with parameters m and g

respectively,  then show that    Z =  ~ b (m,g).

17. Find the m.g.f of Normal distribution.
18. Show that the conditional mean of Y given X is linear in X in the case of bivariate normal distribution.

 

 

 SECTION – C

Answer any TWO questions.                                                   (2 ´ 20 = 40 Marks)

19. Let X₁and X₂ be random variables having the joint p.d.f

Show that the conditional means are

 

(10+10)

20. If f (X,Y) has a trinomial distribution, show that the correlations between

X and Y is   .

 

21. i)    Derive  the p.d.f of ‘t’ distribution with ‘n’ d.f
22. ii) Find all odd order moments of Normal distribution.                       (15+5)
23. i) Derive the p.d.f of ‘F’ variate with (n₁,n₂) d.f                                     (14)

 

1. ii) Define   i)   Null and alternative Hypotheses                                      (2)
2. ii) Type I and Type II errors.                                                (2)

and         iii)   critical region                                                                   (2)

 

 

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS, PHYSICS & CHEMISTRY

AC 10

FOURTH SEMESTER – APRIL 2006

                                                 ST 4201 – MATHEMATICAL STATISTICS

(Also equivalent to STA  201)

 

 

Date & Time : 22-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Part A

Answer all the questions.

1. Define conditional probability of the event A given that the event B has happened.
2. If A₁ and A₂ are independent events with P(A₁) = 0.6 and  P(A₂) = 0.3, find     P(A₁ U A₂), and P(A₁ U A₂^c)
3. State any two properties of a distribution function.
4. Define the covariance of any two random variables X and Y. What happens when they are independent?
5. The M.G.F of a random variable is  [(2/3) + (1/3) e^t]⁵ . Write the mean and variance.
6. Define a random sample.
7. Explain the likelihood function.
8. Let X have the p.d.f. f(x) =1/3, -1<x<2, zero elsewhere. Find the M.G.F.
9. Define measures of skewness and kurtosis through moments.
10. Define a sampling distribution.

 

Part B

Answer any five questions.

1. Stat and prove Bayes theorem.
2. Derive the mean and variance of Gamma distribution.
3. Let the random variables X and Y have the joint pdf

x + y, 0<x<1, 0<y<1

f(x, y) =

0, otherwise.

Find the correlation coefficient.

1. A bowl contains 16 chips of which 6 are red, 7 are white and 3 are blue. If 4 chips are taken at random and without replacement, find the probability that

1. All the 4 are red.
2. None of the 4 is red.

• There is atleast one of each colour.

1. State and prove the addition theorem for three events A, B and C. What happens when they are mutually exclusive?
2. Derive the mgf of Poisson distribution. And hence prove the additive property of the Poisson distribution.
3. Let X₁ and X₂ denote a random sample of size 2 from a distribution that is       N(m, s²). Let Y₁ = X₁ + X₂ , Y₂ = X₁ – X_2.  Find the joint pdf of Y₁ and Y₂ and show that Y₁ and Y₂ are independent.
4. Define the cumulative distribution function F(x) of a random variable X and mention the properties of it.

Part C

Answer any two questions.

19. a) Derive the recurrence relation for the central moments of Binomial distribution. Obtain the first four moments.
20. b) Show that Binomial distribution tends to poisson distribution under certain conditions.           (10 +10 = 20)
21. a) Discuss the properties of Normal distribution
22. b) In a distribution exactly normal, 10.03% of the items are under 25 kilogram weight and 89.97 % of the items are under 70 kilogram weight. What are the mean and standard deviation of the distribution?                                                                                                      (10 +10 = 20)
23. Let f(x, y) = 8xy, 0<x<y<1; f(x, y) = 0 elsewhere. Find
24. a) E(Y/X = x),    b). Var( Y/X = x).
25. b) If X and Y are independent Gamma variates with parameters m and v respectively, show that the variables U = X + Y, Z = X / (X + Y) are independent and that U  is a g( m + v) variate and Z is a b₁(m, v) variate.                                                                                      (10 +10 = 20)
26. a) Derive the pdf of t-distribution.
27. b) Obtain the Maximum Likelihood Estimators of m and s² for Normal distribution.         (10 + 10 = 20)

 

 

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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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       LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

YB 15

FOURTH SEMESTER – April 2009

ST 4206/ ST 4201 – MATHEMATICAL STATISTICS

 

 

 

Date & Time: 27/04/2009 / 1:00 – 4:00  Dept. No.                                                        Max. : 100 Marks

 

 

                                                                        PART – A 

                                                     

ANSWER ALL THE QUESTIONS                                                                          (10 x 2 = 20)

 

1. Define Mutually Exclusive Events with examples.
2. State multiplication law of probability
3. What is the chance that a leap year selected at random will contain 53 sundays?
4. Define probability generating function of a random variable.
5. If X is a random variable and a and b are constants, then show that

E (aX + b) = a E(X) + b provided all the expectations exist.

6. Derive the moment generating function of Poisson distribution.
7. Define Beta Distribution of First kind.
8. Define Regression.
9. Ten unbiased coins are tossed simultaneously. Find the probability of getting at least seven heads.
10. Define Most Powerful test.

 

PART – B

ANSWER ANY FIVE QUESTIONS                                                                          (5 x 8 = 40)

11. Four cards are drawn at random from a pack of 52 cards. Find the probability that
12. These are a king, a queen, a jack and an ace.
13. Two are kings and two are queens
14. Two are black and two are red.
15. There are two cards of hearts and two cards of diamonds.

 

12. The contents of urns I, II and III are as follows:

Urn I   : 1 White, 2 Black and 3 Red balls

Urn II  : 2 White, 1 Black and 1 Red balls, and

Urn III : 4 White, 5 Black and 3 Red balls

One urn is chosen at random and two balls drawn from it. They happen to be white and              red. What is the probability that they come from Urns I, II or III?

13. Let X be a Continuous random variable with probability density function

 

a). Determine the constant a

b). Compute P( X ≤ 1.5 )

 

14. State and prove Chebyshev’s Inequality.

 

15. Derive the Mean and Variance of Binomial Distribution.

 

16. The joint probability distribution of two random variables X and Y is given by

P ( X = 0, Y = 1) =  , P ( X = 1, Y = -1) = and P ( X = 1, Y = 1) =. Find

1. Marginal distributions of X and Y and
2. Conditional probability distribution of X given Y = 1.

 

17. The following figures show the distribution of digits in numbers chosen at random

from a telephone directory:

Digits	0	1	2	3	4	5	6	7	8	9	Total
Frequency	1026	1107	997	966	1075	933	1107	972	964	853	10,000

 

Test whether the digits may be taken to occur with equal frequency in the directory.

18. Define
    1. Null Hypothesis
    2. Alternative Hypothesis

• Level of Significance

1. Two types of Errors                                                     ( 2 + 2 + 2 + 2 )

 

PART – C

ANSWER ANY TWO QUESTIONS                                                                        (20 x 2 = 40)

19. (a). State and Prove Baye’s Theorem . (10)

 

(b). Three groups of Children contain respectively 3 girls and 1 boy, 2 girls and 2 boys,

and 1 girl and 3 boys. One child is selected at random from each group. Show that

the chance that the three selected consist of 1 girl and 2 boys is  .               (10)

 

20. (a). In four tosses of a coin, let X be the number of heads. Tabulate the 16 possible

outcomes with the corresponding values of X. By simple counting, derive the

probability distribution of X and hence calculate the expected value of X        (10)

 

 

 

(b). A random variable X has the following probability density function:

 

x	0	1	2	3	4	5	6	7
p(x)	0	k	2k	2k	3k	k2	2k2	7k2+k

 

1. Find k
2. Evaluate

• If , find the minimum value of a.

1. Determine the distribution function of X (10)

 

21. (a). State any five properties of Normal Distribution (8)

 

(b). A manufacturer, who produces medicine bottles, finds that 0.1% of the bottles

are defective. The bottles are packed in boxes containing 500 bottles. A drug

manufacturer buys 100 boxes from the producer of bottles. Using Poisson

distribution, find how many boxes will contain :

1. no defective and
2. at least two defectives                                                                             (12)

 

22. (a). If , find
    1. Var (X)
    2. Var (Y)

• r (X,Y) (10)

(b). The mean weekly sales of soap bar in departmental stores was 146.3 bars per store.

After an advertising campaign the mean weekly sales in 22 stores for a typical week

increased to 153.7 and showed a standards deviation of 17.2. Was the advertising

campaign successful?                                                                                           (10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHS & PHYSICS

FOURTH SEMESTER – APRIL 2012

ST 4206/4201 – MATHEMATICAL STATISTICS

 

 

Date : 19-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION A

Answer all the questions.                                                                                          10 X 2 = 20

1. Define random experiment.
2. What are independent events?
3. If Var(X) = 4, find Var(3X + 8).
4. Define continuous uniform distribution and write its mean and variance.

1. State any two applications of t-test.

6. Write down the mean and variance of Binomial distribution.
7. Define exponential distribution.
8. Write any two properties of regression coefficients.
9. What is null hypothesis?
10. Define critical region.

SECTION B

Answer any five questions.                                                                                       5 X 8 = 40

11. An urn contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that
12. Two of the balls drawn are white.
13. One ball of each colour is drawn.

• None is red.

1. At least one is white
2. If p₁ =P(A), p₂ =P(B) and p₃ =P(AÇB), (p₁, p₂,p₃ >0); express the following in terms of p₁, p₂ and p₃. ,  P(A/B),  and
3. A random variable X has the following probability function:

x              :               0              1              2              3              4              5              6              7

p(x)        :               0              k              2k           2k           3k           k²            2k²          7k² + k

Find k, evaluate P(X<6).

14. State any four properties of Distribution function.
15. Find mean and variance of Poisson distribution.
16. Derive the r^th order moments of Rectangular distribution and hence find standard deviation.
17. Obtain the line of regression of Y on X for the following data:

X:            65           66           67           67           68           69           70           72

Y:            67           68           65           68           72           72           69           71

18. What are the steps involved in solving testing of hypothesis problem?

 

(PTO)

 

SECTION C

Answer any two questions.                                                                                       2 X 20 = 40

19. a) State and prove addition theorem of probability.
20. b) Sixty percent of the employees of XYZ Corporation are college graduates. Of these, ten are in sales. What is the probability that
21. An employee selected at random is in sales?
22. An employee selected at random is neither in sales nor a college graduate? (10+5+5)
23. The joint probability density function of a two-dimensional random variable (X,Y) is given by:
24. Verify that whether f(x,y) is a joint p.d.f.
25. Find the marginal density functions of X and of Y
26. Find the conditional density function of Y given X=x and conditional density function of X given Y=y.
27. Check for independence of X and Y.        (5+6+6+3)
28. a) A manufacturer, who produces medicine bottles, finds that 0.1 % of the bottles are defective. The bottles are packed in boxes containing 500 bottles. A drug manufacturer buys 100 boxes from the producer of bottles. Using Poisson distribution, find how many boxes will contain:
29. no defective.
30. atleast two defectives. (5+5)
31. Derive mean and variance of Beta distribution of first kind. (10)
32. Derive the p.d.f. of the F-statistic with (n₁, n₂) degrees of freedom. (20)

 

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