Question

Let a,b be

fa,b (a,b) = (1+ab)/3 0<=a<=1 , 0<=b<=2

0 otherwise

a) Find pdf fa and fb

b) Are a and b both indepent? explain why

Answer 1

a) The marginal distributions for a and b here are obtained as:

$f(a) = \int_{0}^{2}f(a, b) \ db$

$f(a) = \int_{0}^{2} \frac{1 + ab}{3} \ db$

$f(a) = \frac{2 + 0.5a*2^2}{3}$

2 + 2a f(a) 3

This is the required marginal PDF for a here.

$f(b) = \int_{0}^{1}f(a, b) \ da$

$f(b) = \int_{0}^{1} \frac{1 + ab}{3} \ da$

$f(b) = \frac{1 + 0.5b}{3}$

This is the required marginal PDF for b here.

b) We have here:

$f(a)f(b) = \frac{2(1 + a)(1 + 0.5b)}{9}$

but this is not equal to f(a, b)

Therefore the two variables are not independent here.