Question

let x and y be random variables whose joint distributionus uniform over the half-disk: {(X,Y)|x^2+y^2 <= 1 and x>0}.

what is the marginal density function of X for 0 <= x <=1?

Answer is 4/pi * (1-x^2)^1/2.

can anyone explain why the reciprocal of the area of semicircle is the density function of the f(X,Y)? Thanks

Answer 1

Solution: We know the pdf f(x) of a random variable X satisfies two conditions,

i)

  and ii) or

To make the area of the density one the comes in the denominator. Because the area of the curve over is  

i.e.   .

This is why  the reciprocal of the area of semicircle is the density function of the f(X,Y).