Jump to content

Wikipedia:Reference desk/Archives/Mathematics/2021 February 10

From Wikipedia, the free encyclopedia
Mathematics desk
< February 9 << Jan | February | Mar >> February 11 >
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.


February 10

[edit]

Expected values

[edit]

On the page of Expected value it is stated that if is a random variable defined on a probability space , then the expected value of , denoted by , is defined as the Lebesgue integral . How does the definition for the case when is a random variable with a probability density function of , (in which case the expected value is defined as ) follow from the general definition? Thanks - Abdul Muhsy (talk) 11:00, 10 February 2021 (UTC)[reply]

If a real-valued random variable has a probability density function , it is the derivative of its cumulative distribution function , so (see Probability density function § Absolutely continuous univariate distributions). The probability function can then be equated with (see Random variable § Distribution functions). So .  --Lambiam 15:20, 10 February 2021 (UTC)[reply]
Can you please explain the first equality in your last equation a bit more? Thanks Abdul Muhsy (talk) 18:03, 10 February 2021 (UTC)[reply]
If (see the sentence immediately before the equation), their differentials are also the same.  --Lambiam 22:23, 10 February 2021 (UTC)[reply]
But I don't understand why given that they have different domains (the sigma field and the real numbers respectively), and also why is ! How does the integral on the probability space turn into an integral on the real line? Is it by first considering the pushforward measure, observing the expectations for X and the identity will be equal and then by Radon-Nikodym theorem applied for the pushforward measure and the Lebesgue measure? Which book will contain a detailed proof Abdul Muhsy (talk) 00:46, 11 February 2021 (UTC)[reply]
It is nothing deep. The event space can be taken to be the Borel -algebra; then its distribution is a measure on (see Borel set § Example). Given a probability distribution the -measure of an interval is where the endpoints may be infinite. For an event represented as a set of disjoint intervals, We can also go the other way: which shows that as far as probability distributions on are concerned, there is a one-to-one correspondence between distributions and probability functions. The lower-case variable corresponds to a possible outcome of . If it is all (notationally) a bit confusing, this is because the theory was originally developed independently (and not always in the most general way) for discrete events and for real-valued random variables, and the concept of probability space was developed afterwards to create a uniform framework capable of capturing these and more.  --Lambiam 10:39, 11 February 2021 (UTC)[reply]