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June 12

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Expected value as integral

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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]