Question

1. Le us sup pose thai the joint probability mass function of two discrete random variables X and Y be given by to,Y) = (1/18) ( x + 2 y), x=1,2;y=1,2 (C)Find the marginal pmf of X (i) Find the marginal pmf of Y (ii) Are X and γ independent? (iv) Find E (X) ) # Mean μ (v) Find Var (X). wnere Var (X) E (X2)-p? (vi) Find standard deviation of X.

Answer 1

i)

marginal pmf of X: f(x)= =(1/18)*(x+2*1)+(1/18)*(x+2*2)=(1/18)*(2x+6)

ii)

marginal pmf of Y: f(y)= $\small \sum_{x=1}^{2}f(x,y)$ =(1/18)*(1+2y)+(1/18)*(2+2y)=(1/18)*(4y+3)

iii)

as f(x)*f(y) is not equal to f(x,y) ; X and Y are not independent

iv)

E(X)= $\small \sum_{x=1}^{2}xf(x)$ =1*(1/18)*(2*1+6)+2*(1/18)*(2*2+6)=28/18=14/9

v)

E(X²)= $\small \sum_{x=1}^{2}xf(x)$ =1²*(1/18)*(2*1+6)+2²*(1/18)*(2*2+6)=48/18=24/9

Var(X)=E(X²)-(E(X))² =24/9-196/81=20/81

vi)

standard deviation of X =SD(X)=sqrt(Var(X))=sqrt(20/81)=0.4969