darth vader approach

Darth Vader was a survivor, and so this approach, which employs the “survival function” will be referred to as The Darth Vader Approach.

The survival function is what is left over from your probabilities once you have summed up the probabilities up to a given value of your random variable X = x. In this way, we can define the survival function as follows:

The Darth Vader Approach utilizes the survival function to find the expected value of a random variable by simply summing from zero to infinity, and it works in the discrete case:

…and in the continuous case:

…and in mixed cases, we simply add the discrete and continuous portions together.

What is the advantage of this approach?

It is a short-cut to the expected value, especially in cases where the cumulative distribution function F(x) is known, but the probability density function, f(x) is not known.