Question

5. Random variables X and Y have joint probability mass function otherwise (a) Find the value of the constant c. (b) Find and sketch the marginal probability mass function Py (u). (c) Find and sketch the marginal probability mass function Px (rk). (d) Find P(Y <X). (e) Find P(Y X) (g) Are X and Y independent? 2 内?

Answer 1

a)

following table is |xi + yj|

		x
		-1	0	1
y	-3	4	3	2
	0	1	0	1
	3	2	3	4
				sum	20

hence 20c = 1

c = 1/20

b)

joint probability mass function

`		x
		-1	0	1
y	-3	0.2	0.15	0.1	0.45
	0	0.05	0	0.05	0.1
	3	0.1	0.15	0.2	0.45
		0.35	0.3	0.35	1

y	p(y)
-3	0.45
0	0.1
3	0.45

c)

x	P(x)
-1	0.35
0	0.3
1	0.35

d)

P(Y < X)

= P(-1,-3)+P(0,-3)+P(1,-3) + P(0,1)       {P(X =x, Y =y) = P(x,y)}

= 0.2 + 0.15 + 0.1 + 0.05

=0.5

e)

P(Y = X)

= P(0,0) = 0