The problem asks to show that for a non-negative random variable $X$, the expected value $\mathbb{E}[X]$ can be expressed as the following integral:

$\mathbb{E}[X] = \int_0^\infty \mathbb{P}(X \geq v) \, dv$

Proof Outline:

We will use the layer cake representation of the expected value, which relies on the fact that for a non-negative random variable, the expected value can be written as an integral over the probability distribution of $X$.

1. Starting with the definition of expectation: $\mathbb{E}[X] = \int_0^\infty x \, f_X(x) \, dx$ where $f_X(x)$ is the probability density function (PDF) of $X$.

2. Using an alternative expression for $x$: We express $x$ as: $x = \int_0^x 1 \, dv$ This allows us to rewrite the expectation as: $\mathbb{E}[X] = \int_0^\infty \left( \int_0^x 1 \, dv \right) f_X(x) \, dx$

3. Changing the order of integration: By Fubini's theorem, we can swap the order of integration: $\mathbb{E}[X] = \int_0^\infty \left( \int_v^\infty f_X(x) \, dx \right) dv$ Notice that the inner integral is the probability that $X$ is greater than or equal to $v$, i.e., $\int_v^\infty f_X(x) \, dx = \mathbb{P}(X \geq v)$

4. Final expression: Thus, we obtain: $\mathbb{E}[X] = \int_0^\infty \mathbb{P}(X \geq v) \, dv$ which is the desired result.

This proof shows that the expectation of a non-negative random variable can be calculated by integrating the tail probability $\mathbb{P}(X \geq v)$ over $v$.

Let me know if you need further clarifications or details. Here are some related questions:

1. How does this result relate to the expectation of other types of random variables?
2. Can this integral form of the expectation be used for discrete random variables?
3. How does this result apply to common distributions like the exponential or uniform distributions?
4. What are the conditions required for Fubini’s theorem to allow the exchange of integrals?
5. How would this change if the random variable $X$ is not non-negative?

Tip: The tail probability $\mathbb{P}(X \geq v)$ is crucial in understanding the behavior of random variables in risk theory and survival analysis.