Question

The joint probability mass function of random variables X and Y is given by if x1 = 1,2; x2 = 1,2 p(x1, x2) = { otherwise (a) Specify the probability mass function of X1 and X2. (b) Are X1 and X2 independent? Are they identically distributed? Explain. (C) Find the probability of the event that X1 + 2X2 > 3. (d) Find the probability of the event that X1 X2 > 2.

Answer 1

a)

p(X1) =$\sum_{x2=1}^{x2=2} p(x1,x2)$ =(x1*1+1)/13+(x1*2+1)/13 =(3x1+2)/13 for x1=1,2

similarly p(x2)=(3x2+2)/13 for x2=1,2

b)

No They are not independent as P(x1,x2) is not equal to P(x1)*P(x2)

Yes they are identically distributed as there pmf is same

c)

P(X1+2X2>=3) =P(X1=1,X2=1)+P(X1=1,X2=2)+P(X1=2,X2=1)+P(X1=2,X2=2) =1 (As it contain all the sample space for minimum value of X1+2X2 is 3 for sample space)

d)P(X1X2>2) =P(X1=2,X2=2) =(2*2+1)/13 =5/13