LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 12

THIRD SEMESTER – April 2009

ST 3501 / ST 3500 – STATISTICAL MATHEMATICS – II

 

 

 

Date & Time: 06/05/2009 / 9:00 – 12:00     Dept. No.                                                       Max. : 100 Marks

 

 

 

SECTION A

Answer ALL questions.                                                                 (10 x 2 =20 marks)

 

1. Define upper sum and lower sum of a function.
2. When do you say that a function is Riemann integrable on [a,b]?
3. Define moment generating function of a continuous random variable.
4. Define improper integrals.
5. State the μ test for convergence of integrals.
6. Define Laplace transform of a function.
7. Let f(x) =    cx ², 0 < x < 2

0, otherwise

If this is a probability density function, find c.

1. State the second fundamental theorem of integral calculus.
2. State any two properties of Riemann integral.
3. Define absolute convergence of a function.

 

SECTION B

Answer any FIVE questions.                                                         (5 x 8 =40 marks)

 

1. For any partition P  on [a, b], prove that m (b – a) ≤ L (P, f) ≤ U (P, f) ≤ M (b – a).

Where m =  , M = and f is a bounded function on [a,b].

1. Let f(x) = x ², 0 ≤ x ≤ 1. By considering partitions of the form  P _n =

Show that  U (P_n, f) =  L (P_n, f) = 1/3

1. Let X be a random variable with probability density function

f(x) =   1, 0 < x < 1

0, otherwise

Find the moment generating function and hence the mean and variance of X.

1. Show that   (a > 0)   , converges for p > 1 and diverges for p ≤ 1.
2. Discuss the convergence of the following improper integrals:

(i)                                        (ii)

 

1. Show that  (i)      and      (ii) b (m, n) = b (n, m)
2. Show that L _{(f + g)}(s) = L_f (s) + L_g (s) and L_{c f} (s) = c L_f (s), where L_f (s) is the Laplace transform of the function f.
3. Evaluate   over the region between the line x = y and the parabola y = x ².

 

SECTION C

 

Answer any TWO questions.                                                (2 x 20 =40 marks)

 

1. (a) Let f, g Є R [a, b], then show that f + g Є R [a, b] and

(b) Let f (x) =      5 x ⁴, 0 ≤ x < 1

0, otherwise

be the probability density function of the random variable X.

 

Find (i) P (1/2 < X < 1) (ii) P (-2 < X < 1/2) (iii) P (0 < X < 3/4) (iv) P (1/4 < X < 3)

 

1. (i) State and prove the first fundamental theorem of Integral calculus.

 

(ii) Derive the differential difference equations for a Poisson process.

21. (i) Show that b (m, n) =

      (ii) If X has the probability density function f (x) = c x ² e^-x, 0 < x < . Find

            Expectation of X and variance of X.

 

22. (a) If  converges absolutely, then show thatconverges.

      (b) Discuss the convergence of the following integrals.

(i)    (ii)

 

 

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