Q6.A random variable X has the following probability distribution:

X 0 1 2 3 4 5 6 7

P(X) 0 k 2k 2k 3k k² 2k² 7k²+k

Find (a) k and hence evaluate P(0 < X < 5)

(b) P(1.52).Also find the E(X).

Q7.The joint distribution function of a random variable (X,Y) is given by

F_xy(x,y)= {(1-e^-ax)(1-e^-by) ; x,y>=0, a,b>0

Find (i) Marginal distribution functions of X and Y.

(ii) P(X<=2 , Y<=2 and P(X<=2). Also show that X and Y are independent.

Q8. The joint probability mass function of (X , Y) is given by

P_xy(x_i,y_j)= {k x_i² y_j² ; i=1,2 ;j=1,2,3

(i) Find k.

(ii) Find the marginal probability mass function of X and Y.

Q9. The joint probability density function of a bivariate variable (X, Y) is given by

f_xy (x, y) ={k (x + y) ; 0 < x < 3 , 0 < y < 3

where k is constant.

(i) Find the value of k.

(ii) Find the marginal probability density function of X and Y.

(iii) Are X and Y independent?

Q10.The joint probability density function of a bivariate variable (X, Y) is given by

f_xy (x, y) ={k ( 2 x + y) ; 0 < x < 1 , 0 < y < 1 where k is constant.

(i) Find the value of k.

(ii) Find the marginal probability density function of X and Y.

(iii)Conditional density of X for given Y and use it to evaluate P (X<=1/2 / Y=1).