Suppose that X is randomly drawn from a uniform distribution

Suppose that X is randomly drawn from a uniform distribution on the interval [0, 2]. Also, suppose that alter the value X = x has been observed (0

Solution

In E(Y|x) we are calculating the conditional expectation for Y given that X is observed as x and
Y is limit by x as lower limit of Y from uniform distibution is dependent of x.

Whereas in E(Y|X) we are calculating the conditional expectation for Y given X = x but in this case Y is not limit by x as its limilt is independent of x