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2 Marks

1. If Var(X) = 4, find var(3X+8) , where X is a random variable. (A.U, Model)

2. X and Y are independent random variables with variance 2 and 3 . Find the variance of 3X +

4Y.[A.U. April/May ,2003]3. Let X be a R,V with E(X) = 1, and E(X(X-1)] = 4. Find Var X and Var(2-3X).[A.U.

May,2000]4. Let X be a R.V taking values 1 , 0 and 1 such that P(X = -1 ) = 2P(X=0) =P(X = 1) . Find themean of 2X-5.

5. A continuous random variable X has probability density function given by f(x) = 3x2, 0 X

1. Find K such that P(X > K ) = 0.05(A.U. Model )

6. The p.d.f of a random variable X is f(x) = 2x, 0 < x < 1, find the p.d.f of Y = 3X + 1.[A.U.A/M 2003]

7. A random variable X has the p.d.f f(x) given by

f(x) = C X e-x, If X > 0

0, If X 0

Find the value of C and c.d.f of X8. The first four moments of a distribution about X 4 are 1, 4 10 and 45

4 = 3 = 0 and Respectively . Show that the mean is 5 , variance is 3, 26.[A.U.mN/D 2004

2 . Find the first two8. For a binomial distribution mean is 6 and S.D is termsof the distribution.[A.U. A/M 2004]

10. Define poisson distribution [A.U. N/D 2005]

11. If X is a poisson vitiate such that P( X = 2) = 9 P( X =4) + 90 P(X =6), findthe variance. A/M 2003]

12. The moment generating function of a random variable is given by

Mx (t) = e3(e 1) . Find P( X =1)[AU, Model]

13. Find the moment generating function of the generating

distribution.[A.U.Dec.98]14. Find the moment generating function of a uniform distribution.[A.U,2000]

15. If the random variable X is uniformly distributed over (-1,1), find the density

x/2) Function of Y = sin(/2) , find the probability /2, 16. If X is uniformly distributed in (- distributionfunction of y = tan x.[A.U.N/D 2003]

17. Define Gamma distribution.[A.U.N/D 2004]

18. Let X be a R.V with p.d.f given byf(x) = 2x, 0 < X < 1

0, elsewhereFind the p.d.f of Y = (3X + 1)[A.U.2000)19.The life time of a component measured in hours follows weibull = 0.5. Find mean lift time of

the = 0.2, distribution with parameter component.[A.U. April 03]

X =10. Find P(15 = 20 and S.D . 20. A normal distribution has mean 40).[ A.U.Nov07 ,May 03]

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16 Marks

1.Let X be a discrete random variable whose cumulative distribution function is [A.U May 2000]

0 for x < -3

6 X F(X) = 1/6 for 310 X 1/2 for 610 1 for x4) , P (-5 (a) Find P( X < 4 ), XP( x = -3 ), P( x= 4 )(b) Find the prob. Mass function of x

2. A random variable X has the following probability function

X: 0 1 2 3 4

P(X) K 3K 5K 7K 9K

(i) Find the value of K

(ii) Find P( X < 3) , P( 0 3 ), P( X < X < 4 )( iii)Find the distribution function of X2. Letr X be a R. V with p.d.f given by

f(x) = 2x, 0 < x < 1

0, elsewhereFind the pdf of Y = (3X + 1) [A.U. 2000]

3.Find the moment generating function of an exponential random variable and hence find its

mean and variance.[ A.U. N/D 2004]

0. fiind 4. A continuous random variable X has the p.d.f f(x) = Kx2 e-x, X the rth moment ofX about the origin. Hence find mean and variance of X. [ A.U. A/M 2003]

5. If X has the probability density f(x) = Ke-3x , X > 0 , fiind K , 1 ) and the mean of X. [A.U.

A/M 2004] X P(0.56. Let the random variable X have the p.d.f

e-x/2f(x) = 0, otherwise

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

7.Define Binomial distribution, obtain its MFG , mean and variance.[A.U.N?D 2003,A/M 2004]

9. The probability of a bomb hitting a target is 1/5 , Two boms are enough to destroy a bridge. Ifsix bombs are aimed at the bridge, find the probability that the bridge is destroyed? [ A.U Dec

98]

10. Describe Binomial B(n,p) distribution and obtain the moment generating function. Hencecompute (i) The first four moment and (ii) the recursion relation for the central moments.

[ A.U.A/M 2005]

11. If X and Y are independent poisson random variables, show that the conditional distributionof X given X+Y is a binomial distribution.[A.U. model]

12. If a random variable X has a negative binomial distribution. Obtain the mean and variance of

X.[A.U. A/M 2004]

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13. Describe negative binomial distribution X follows NB( k,p) where X = number of failures

preceding the kth success in a sequence of Bernoulli trials and p = Probability success. Obtain

the MGF of X, mean and variance of X.14. Let X be a uniform random variable over 90,1) . Determine the moment generating function

of X and hence find variance of X . Given that X is a uniform random variable over (0,1) .

Hence its p.d.f is given by f(x) = 1/1-0 =1 , 0 < x < 1.15. Let Y = X2 , find the p.d.f of Y if X is a uniform random variable over (-1,2).

16. If a poisson variate X is such that P( X =1) = 2P(X = 2). Find P( X = 0) and var(X) . If X is a

uniform random variable in [-2,2] , find the p.d.f of Y = and E[Y].X17. If x is a uniform random variable in the interval (-2,2) find the p.d.f of Y = X2 [A.U.A/M

2005]

18. Find the moment generating function of an exponential random variable and hence find itsmean and variance.[A.U. N/D 2004]

19. If X and Y are independent exponential distributions with parameter 1, find the pdf of U = X

Y. [A.U. Model]

20. The daily consumption of milk in excess of 20,000 gallons is = 3000 . The city has a

daily approximately exponentially distributed with stock of 35,000 gallons. What is theprobability that of two days selected at random, the stock is insufficient for both days.[ A.U.A/M2003]21. Obtain the moment generating funtion of a Gamma variable X. Hence or otherwise calculate

the mean and variance of X.