Calculating Expected Value from CDF

Question

I have two questions if someone can help me and give me reference used.

Can we calculate Expected Value (EV) by reading random variables from Cumulative Distribution Function (CDF)? For example, P90 = 467, P50 = 740 and P10 = 1,139? Can the EV be 90%*467 + 50% * 740 and 10% * 1,139.

Can the sum of probabilities be more than 100% when calculating EV? I see the following calculations.

EV = 97% * 467 + 70% *273 + 30% * 399 = 764

The author claims that 764 will be close to P50 = 740 if the data is represented by normal probability distribution.

Answer 1

Another generic connection between the cdf $F$ and the mean $\mathbb E[X]$ is given by the identity $$\mathbb E[X]= \int^{-\infty}_0 F(x)\,\text dx+\int^{\infty}_0 (1-F)(x)\,\text dx$$ which appears in many X Validated entries, e.g.

• Expectation when cumulative distribution function is given
• Does a univariate random variable's mean always equal the integral of its quantile function?

Answer 2

If you have the analytical formula for CDF, then yes, you can calculate EV from that (one way to do this is to take the derivative of CDF to drive PDF and then use the integral formula for EV).

Here, you only have three points on CDF. This means you can only approximate EV (unless you are sure that your random variable only takes these three particular values). Your example cannot be CDF as CDF is the probability that the variable takes a value less than or equal to x, so its output should be between zero and one. What you have is probably 1-inversCDF. If that's the case, P90=467 tells you 90% of the population lies after 467 (equivalently, 10% is less than 467). By the same token, from P90=467 and P50=740, we can infer that 40% of the population lies between 467 and 740. In this case, one approximation to EV can be 10%*467 + 40% * 740 + 50% * 1,139