Answer to Question #215090 in Statistics and Probability for Arshad Ali Wazir

Question

Q.3 : Let X, Y be two random numbers with joint distribution function 

f(x,y) = Kx fory≤x≤y+1,0≤y≤2, 

= 0 otherwise. 

(a) Compute K. 

(b) Compute the m.d.f. and the expectation of X. 

Answer 1

1= $\intop$ ^+ϖ $\intop$ _-ϖ ⁺ϖ f(x,y)dxdy =$\int$ ₀²($\int$ _y^y+1kxdx)dy= k/2$\intop$ ₀² ((y+1)² -y)dy= k$\intop$ ₀² (y+ 0.5)dy

k/2(y+0.5)² from 0 to 2

= k/2 (2.5² -0.5²) =3k

Therefore k =1/3

Mean

E(x) =$\iint$ xf(x,y)dxdy =1/9$\intop$₀²((y+1)³-y³)dy = 1/36(y+1)⁴ -y⁴) from 0 to 2

= (3⁴ -2⁴ -1⁴ +0)/36

=16/9

The probability density function of x is f_x(x) =$\intop$ f(x,y)dy.

If x ∈/ (0,3)

f_x(x)= 0 otherwise

f_x(x) = $\intop$ _{(0, x-1)}^(x,2) x/3.