Question

Problem 1.20 (Integration by parts) Prove that E[X]=∫ 0 [infinity] ​ (1−F X ​ (x)dx−∫ −[infinity] 0 ​ F X ​ (x)dx whenever at least one of the integrals is finite.

Answer 1

To prove the equation E[X]=∫0∞ (1−FX(x))dx−∫−∞0 FX(x)dx, we use integration by parts and assume X is a continuous random variable with probability distribution function FX(x).

Step-by-step explanation:

To prove that E[X]=∫0∞ (1−FX(x))dx−∫−∞0 FX(x)dx holds whenever at least one of the integrals is finite, we can use integration by parts.

Let's assume that the random variable X is continuous with a probability distribution function FX(x). Using the formula for integration by parts, we can write:

∫abu dv = uv−∫abv du

In this case, we can set u = (1−FX(x)) and dv = dx. Therefore, du = -fX(x) dx and v = x.

By substituting these values into the integration by parts formula and evaluating the integrals, we can prove the given equation.