Wikipedia:Reference desk/Archives/Mathematics/2023 June 12

Mathematics desk
< June 11 << May | June | Jul >> June 13 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


June 12

edit

Expected value as integral

edit

For a non-professional, why exactly is the expected value defined as  , as is stated here? Ideally, explain step by step please.--Hildeoc (talk) 22:09, 12 June 2023 (UTC)[reply]

If the probability density function of a random variable   is given as function on the reals, we can approximate its expected value by partitioning the real number line   into an infinite collection of small intervals   The probability that an outcome of   falls in the interval   is   Abbreviating this as   the approximation is then given by the sum  
Now recall that  
You may see where this is going. The value of   lies between the lower and upper Darboux sums for   for that same partitioning, so as the partitioning gets arbitrarily fine, the approximation tends to the value of that integral.  --Lambiam 05:14, 13 June 2023 (UTC)[reply]