February 10 Information

On the page of Expected value it is stated that if ${\displaystyle X}$ is a random variable defined on a probability space ${\displaystyle (\Omega ,\Sigma ,\operatorname {P} )}$, then the expected value of ${\displaystyle X}$, denoted by ${\displaystyle \operatorname {E} [X]}$, is defined as the Lebesgue integral ${\displaystyle \operatorname {E} [X]=\int _{\Omega }X(\omega )\,d\operatorname {P} (\omega ).}$. How does the definition for the case when ${\displaystyle X}$ is a random variable with a probability density function of ${\displaystyle f(x)}$, (in which case the expected value is defined as ${\displaystyle \operatorname {E} [X]=\int _{\mathbb {R} }xf(x)\,dx}$) 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 ${\displaystyle f}$, it is the derivative of its cumulative distribution function ${\displaystyle F}$, so ${\displaystyle \int _{S}y\,dF=\int _{S}yf(x)\,dx}$ (see Probability density function § Absolutely continuous univariate distributions). The probability function ${\displaystyle \operatorname {P} }$ can then be equated with ${\displaystyle F}$ (see Random variable § Distribution functions). So ${\displaystyle \int _{\mathbb {R} }x\,d\operatorname {P} =\int _{\mathbb {R} }x\,dF=\int _{\mathbb {R} }xf(x)\,dx}$.  -- 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 ${\displaystyle \operatorname {P} =F}$ (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 ${\displaystyle \operatorname {P} =F}$ given that they have different domains (the sigma field and the real numbers respectively), and also why is ${\displaystyle \operatorname {E} [X]=\int _{\Omega }X\,d\operatorname {P} =\int _{\mathbb {R} }x\,d\operatorname {P} .}$! 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 ${\displaystyle {\mathcal {F}}}$ can be taken to be the Borel ${\displaystyle \sigma }$-algebra; then its distribution is a measure on ${\displaystyle {\mathcal {F}}}$ (see Borel set § Example). Given a probability distribution ${\displaystyle F,}$ the ${\displaystyle F}$-measure of an interval ${\displaystyle I=[x_{0},x_{1}]}$ is ${\displaystyle F(I)=F(x_{1})-F(x_{0}),}$ where the endpoints may be infinite. For an event ${\displaystyle {\mathcal {I}}}$ represented as a set of disjoint intervals, ${\textstyle \operatorname {P} ({\mathcal {I}})=\sum _{I\in {\mathcal {I}}}F(I).}$ We can also go the other way: ${\displaystyle F(x)=\operatorname {P} (\{[-\infty ,x]\}),}$ which shows that as far as probability distributions on ${\displaystyle \mathbb {R} }$ are concerned, there is a one-to-one correspondence between distributions and probability functions. The lower-case variable ${\displaystyle x}$ corresponds to a possible outcome of ${\displaystyle X}$. 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