91Ó°ÊÓ

The joint distribution of X and Y is a uniform distribution with the circle \({x^2} + {y^2} < 1\)

The joint p.d.f. of X and Y is,

\(f\left( {x,y} \right) = \left\{ {\begin{align}{}c&{for\,\,{x^2} + {y^2} < 1}\\0&{otherwise.}\end{align}} \right.\)

Therefore, for any given value of y in the interval\( - 1 < y < 1\)

The conditional p.d.f. of X given that\(Y = y\)is,

\(\begin{array}g\left( {x\left| y \right.} \right) = \frac{{f\left( {x,y} \right)}}{{f\left( y \right)}}\\ = \left\{ {\begin{array}{}{\frac{c}{{f\left( y \right)}}}&{for\,\, - \sqrt {1 - {y^2}} < x < \sqrt {1 - {y^2}} }\\0&{otherwise}\end{array}} \right.\end{array}\)

For each given value of y, this conditional p.d.f. is a constant throughout values of x symmetric concerning \(x = 0\)

Therefore,\(E\left( {X\left| Y \right.} \right) = 0\)for each value of y