# Sum of estimated costs for uncertain events

I have a number of possible events $$e$$ with a probability $$p_e$$ of the event occuring and a cost estimate should the event occur (if it doesn't occur the cost is 0). The probability for each event is a known and so is the mean cost $$c_e$$ and standard deviation $$\sigma_e$$ for the cost of each event should it occur.

How do I calculate the expected value and standard distribution for the sum of these events (the total cost expected)? Assume the events are independent and normal distributed.

You can sum expectations and for independent random variables can sum variances. You do not need a normal distribution assumption.

If an event occurs, then its cost has mean $$c_e$$ and variance $$\sigma^2_e$$ so the expectation of its square is $$\sigma^2_e+c_e^2$$

Since the event occurs with probability $$p_e$$, taking this into account its expected cost is $$p_ec_e$$ and the expectation of its square is $$p_e\sigma^2_e+p_ec_e^2$$, so its variance is $$p_e\sigma^2_e+p_ec_e^2 - p_e^2c_e^2$$

So in sum, the total cost has:

• expectation $$\sum\limits_e p_ec_e$$
• variance $${\sum\limits_e \left(p_e\sigma^2_e+p_ec_e^2 - p_e^2c_e^2\right) }$$
• standard deviation $$\sqrt{\sum\limits_e \left(p_e\sigma^2_e+p_ec_e^2 - p_e^2c_e^2\right) }$$
• Thank you! Took me a while to understand it (I've always been slow to get statistics). – nononono Feb 12 at 18:48