SSN COLLEGE OF ENGINEERING, KALAVAKKAM

DEPARTMENT OF MATHEMATICS  

ASSIGNMENT ON RANDOM VARIABLES  

PART A  

1.   If two random variables X and Y have probability density function (PDF)

for x, y>0, evaluate ‘k’

 

2.   Let X and Y be integer valued random variables with

Are X and Y independent?

3.   The p.d.f. of a random variable X is f(x) =2 x , 0 <  x <1,  find the p.d.f.  of Y=3X+1.

4.   If  X and Y are random variables having the joint density function

5.   Can the joint distributions of two random variables X and Y be got if their marginal distributions are known?

6.   State the basic properties of joint distribution of (X, Y) where X and Y are random variables

7.   If X and Y have joint pdf check whether X and Y are independent.

8.   Is the function defined as follows a density function?

9.   Find the marginal density function of X and Y if .

10.                    A continuous random variable X has the p.d.f.  f(x) given by f(x)=

  Find the value of C and C.D.F. of X.

11.                    If the joint p.d.f. of (X, Y) is f(x, y) = find

12.                    The  following table gives the joint probability distribution of X and Y.  Find the marginal density functions of  X and  Y.

X

Y

1 2 3

1 0.1 0.1 0.2

2 0.2 0.3 0.1

13.                    If the joint pdf of the random variables (X, Y) is given by

find the value of k.

14.                    Find k if the joint probability density; function of a bivariate

random variable  (X, Y) is given by

 

 

 

15.                    Two random variable X and Y has joint probability density function

Find E (XY).

16.                    The joint probability mass function of (X, Y) is given by P (x, y) = K (2x + 3y),   x = 0, 1, 2;  y= 1, 2, 3.  Find the marginal probability distribution of

X :

 

  Y Y Y

X 1 2 3

0 3K 6K 9K

1 5K 8K 11K

2 7K 10K 13K

17.                    A continuous random variable X that can assume any value between x=2 and  x=5 has a density function given by; f(x) = k (1+x). Find P [X<4].

18.                    A random variable X has to p.d.f. f(x) given by                    

Find the value of C and C.D.F. of X.

19.                    If the joint p.d.f. of (X, Y) is given by

Find           E ( XY).

20.                    Find the value of (a) C and (b) mean of the following distribution

given  

         

PART B  

1.   The joint probability mass function of random variables X and Y is

are constants.  Find the marginal and conditional distributions. 2.   Let the number X be selected from among the set of integers {1, 2, 3, 4}and the number Y be chosen from among those at least as X.  Obtain the joint PMF of (X, Y).  Hence prove that covariance (X, Y) =5/8. 3.   Two dimensional random variable (X, Y) has the joint PDF

otherwise Find (1) marginal and conditional distributions, and (2) Test whether X and Y are independent.

4.   If  X has the probability density   find

and the mean of X.

5.   If the joint density of is given by

find the probability density of

and its mean. 6.   Random variables X and Y have the joint distribution

Find the marginal and conditional distributions and evaluate P{X<1}.

7.   Suppose the point probability density function (PDF) is given by

.  Obtain the marginal PDF of  X and

that of Y. Hence or otherwise find 8.   The joint probability mass function (PMF) of X and Y is

   

P(x, y) 0 1 2

0  

1  

X       2    

0.1 .04 .02

.08 .20 .06

.06 .14 .30

Compute the marginal PMF  X and of Y, and check if X and Y are independent.

9.   The joint density function of random variables X and Y is find the marginal and conditional probability density

functions. Are X and Y independent?

10. The joint of X and Y is given by find

the probability density function of 11. If the joint pdf of a two dimensional random variable (X, Y) is given by

Find (i) (ii)  

                     (iii) . 12. If X is the proportion of persons who will respond to one kind of mail order solicitation, Y is the proportion of persons who will respond to another kind of mail-order solicitation and the joint probability density of X and Y is

given by Find the Probabilities that (i) at least 30% will respond to the first kind of mail-order solicitation.(ii) atmost 50% will respond  to the second kind of mail-order solicitation given that there has been 20% response to the first kind of mail-order solicitation.

13. Suppose the probability density function is given by

.  Obtain the marginal of X, the conditional pdf of Y given X = 0.8 and then E(Y/x=0.8). 14. Given is the joint distribution X and Y:

    X X X X X 0 1 1 2 2

  Y

0 1 2

0.02 0.05 0.03

0.08 0.02 0.20 0.12

0.05 0.03

0.10 0.08 0.25 0.15

0.20 0.12

Obtain Marginal Distributions and the conditional distribution of X given         Y = 0.                          

15. Given the joint pdf of (X, Y) as  16. Find the marginal and conditional pdfs  of X and Y.  Are  X and Y independent?

  17. The joint pdf of random variable X and Y is given by

.  Find the value of K and prove also that X and Y are independent. 18. The joint p.d.f. of a bivariate R.V. (X, Y) is given

by:                                        f(x, y) = Where k is a constant (1) Find the value of K (2) Find P(X+Y<1) (3) Are X and Y independent random variables.  Explain. 19. Let X and Y be independent standard normal random variables.  Find the

p.d.f. of    

20. If the Joint p.d.f. of random variables X and Y  is                                                                               f(x, y) =

find f(y/x) and E(Y/X=x).                                     21. Let X and Y be independent uniform random variables over (0, 1).Find the p.d.f. of Z=X+Y

22. If the joint density function of the two random variables X and Y be    

Find : (1)  

23. Given Find (1)C, (2) The marginal distributions  f(x) and  f(y) and (3)The conditional density of Y given X f(y/x) 24. If X and Y each follow an exponential distribution with parameter 1 and are independent, find the pdf of U=X-Y. 25. The diameter of an electric cable X is a continuous random variable with pdf f(x)=kx(1-x), 0   Find (A) the value of k (B) the cumulative

distribution function of X (C) P (X

26. If X and Y are independent random variable with and

  find the density function of Are they independent? 27. If the joint pdf of a random variable (X, Y) is given by

find the conditional densities of X given Y and Y given X.             

 

28. The pdf of  X and Y is given by 29. Find k and prove that X and Y are independent

  30. X and Y are two random variables having joint density function

                                                             Find

31. Two random variables X and Y have the following joint probability

density function f(x, y) =    Find   (1) Marginal probability density functions of X and Y  (2) Conditional density functions  (3) var (X) and var (Y). 32. Let (X, Y) be a two-dimensional non-negative continuous random

variable having the joint density.              

Find the density function of 

33. The joint p.d.f. of R.V.s X and Yis given by

Find the marginal p.d.f. of X , P(X+Y < ½),Cov (x, Y). 34. The joint p.d.f. of R. vs X and Y is given by                                              

f(x) =              35. The random variables X and Y have joint p.d.f.

Are X and Y independent? Find the conditional p.d.f. of X given Y.

  36. Suppose X and Y are two random variables having the joint p.d.f.

. Find the p.d.f of  

37. In producing gallium – arsenide microchips, it is known that the ratio between gallium and arsenide is independent of producing a high percentage of workable wafer, which are main components of microchips.  Let X denote the ratio of gallium to arsenide and Y denote the percentage of   workable mierowafers retrieved during a 1-hour period. X and Y are independent random variables with the joint density; being known as

Show that E (XY)=E(X). E(Y).

38. If the joint density of given by

find the probability density of    

39. Two random variables X and Y have joint density function

Find the conditional density functions.  Check whether the conditional density functions are valid

 

40. If the joint probability density of

find the probability of