Question

Consider the random variables X and Y with joint density function ???? f(x,y)= x+y, 0≤x≤1;0≤y≤1 0, elsewhere. (a) Find the marginal distributions of X and Y . (b) Find P(X > 0.25,Y > 0.5).

Answer 1

a. The marginal densities

$f_X(x)=\displaystyle\int_0^1(x+y)\,\mathrm dy=x+\frac12$

and

$f_Y(y)=\displaystyle\int_0^1(x+y)\,\mathrm dx=y+\frac12$

b. This can be obtained by integrating the joint density over [0.25, 1] x [0.5, 1]:

$P(X>0.25,Y>0.5)=\displaystyle\int_{1/4}^1\int_{1/2}^1(x+y)\,\mathrm dx\,\mathrm dy=\frac{33}{64}$