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.


1 Answer 1


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 $\sqrt{\sum\limits_e \left(p_e\sigma^2_e+p_ec_e^2 - p_e^2c_e^2\right) }$
  • $\begingroup$ Thank you! Took me a while to understand it (I've always been slow to get statistics). $\endgroup$
    – nononono
    Commented Feb 12, 2021 at 18:48

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