Question

Let the joint pmf of X and Y be p(x, у) схуг, x-1,2,3, y-12. a) Find constant c that makes p(x, y) a valid joint pmf. c) Are X and Y independent? Justify d) Find P(X+Y> 3) and PCIX-YI # 1)

Answer 1

a)

for this to be valid: $\sum_{x=1}^{3}$ $\sum_{y=1}^{2}p(x,y)$=1

$\sum_{x=1}^{3}$$\sum_{y=1}^{2}p(x,y)$ =c(1*1²+2*1²+3*1²+1*2²+2*2²+3*2²)=c*30=1

c=1/30

b)

marginal pmf of X: f(x)=$\sum_{y=1}^{2}p(x,y)$ =(1/30)*(x*1²+x*2²)=5x/30 =x/6 for x=1,2,3

marginal pmf of y: f(y)=$\sum_{x=1}^{3}p(x,y)$ =(1/30)*(1*y²+2*y²+3*y²)=6y²/30 =y²/5 for y=1,2

c)

as P(x,y)=P(x)*P(y) ; therefore x and Y are independent

d)

P(X+Y>3)=P(X=2,Y=2)+P(X=3,Y=1)+P(X=3,Y=2)=(1/30)*(2*2²+3*1²+3*2²)=23/30

P(|X-Y|$\neq$1)

=P(X=1,Y=1)+P(X=2,Y=2)+P(X=3,Y=1)=(1/30)*(1*1²+2*2²+3*1²)=12/30=2/5