# Proof of the total probability rule for expected value?

1. Feb 1, 2017

### theone

1. The problem statement, all variables and given/known data
Does anyone know of a simple proof for this: https://s30.postimg.org/tw9cjym9t/expect.png

E(X) = E(X|S)P(S) + E(X|S_c)P(S_c)

X is a random variable,
S is an a scenario that affects the likelihood of X. So P(S) is the probability of the scenario occurring and and P(S_c) is the probability of the scenario not occurring

2. Relevant equations

3. The attempt at a solution

2. Feb 1, 2017

### PeroK

How do you define $E(X)$?

3. Feb 1, 2017

### theone

the expected value of the random variable X; the probability weighted average of the possible outcomes of X

4. Feb 1, 2017

### PeroK

You can't prove anything with just words. You need a mathematical definition.

5. Feb 1, 2017

### theone

Do you mean this:
$E(X) = \sum_{i} X_i P(X_i)$

6. Feb 1, 2017

### PeroK

If you have some properties for $P$ you could take it from there.

7. Feb 1, 2017

### xtempore

Not sure about a mathematical proof, but doesn't that formula just state the obvious? Perhaps putting it into words makes it clearer.

The expected value of X is the sum of the expected value of X when S happens multiplied by the probability that S happens plus the expected value of X when S doesn't happen times the probability of S not happening.

Because S happening and S not happening are mutually exclusive you can just add the two values together.

For a mathematical proof, you'd probably want to include your definition of E(X), the fact that P(S) + P(S') = 1, and the basic conditional probability formula (https://en.wikipedia.org/wiki/Conditional_probability)

Then go from there.

8. Feb 1, 2017

### Ray Vickson

You need formulas for $E(X|S)$ and $E(X|S_c)$. Do you know what they are?