I need some help with the following problem:
Let X be a continuos random variable that takes values from a to b, where a and b are finite. X follows a given distribution Function 'F'. Suppose that g(X) is a continuous and differentiable function. E.g., g=k*X, where k is a positive constant.
I am trying to compute E(g(X)|X<=x0), where x0<b. I know X<=x0 because a given event has occurred that excludes some interval of values from the support of the original distribution function.
In the example above, is the solution the integral from a to x0 of g(x).f(x)?
$$f_{X|X<x_0}(t)=\frac{1}{F_X(x_0)}f_X(t)\mathbb{1}_{[a;x_0]}(t)$$
thus your expctation becomes
$$\mathbb{E}[g(X)|X<x_0]=\frac{1}{F_X(x_0)}\int_{a}^{x_0}g(x)f_X(x)dx$$