Question

Questionl The random variable X and Y have the following joint probability mass function: 0.14 0.27 0.2 0.1 0.03 0.15 0.1 a) Determine the b) Find P(X-Y>2). c) Find PX s3|Y20) d) Determine E(XY) e) Determine E(X) and E(Y). f) Are X and Y independent? marginal pmf for X and Y. Question 2 Let X and Y be independent random variables with pdf 2-y 0sxS 2 f(x)- f(p)- 0, otherwise 0, otherwise a) b) Find E(XY). Find Var (2X + 3Y).

Answer 1

1-a)

Following table shows the marginal pdfs of X and Y:

			X
		2	3	4	P(Y=y)
	-1	0.1	0.2	0.05	0.35
Y	0	0.27	0.1	0.15	0.52
	1	0	0.03	0.1	0.13
	P(X=x)	0.37	0.33	0.3	1

b)

c)

So the required probability is

=-= 200.650.6154

d)

Following table shows the calculations:

e)

Following table shows the calculations:

P(X-x)Y 0.37 0.33 0.3 -1 0.35 0.52 0.13 0.74 0.99 1.2 2.93 0.35 0 0.13 0.22 0 4 Total

So we have

y) =-0.22

f)

Since P(Y=-1) = 0.35, P(X=2) = 0.37 and P(X=2,Y=-1) = 0.1 so P(X=2, Y=-1) is not equal to P(X=2)*P(Y=-1). Therefore X and Y are not independent.