"My friend hasn't / hadn't been in church in two weeks". For example, if $\mathcal G$ is generated by two real-valued random variables $Y, Z$, then $E[X] = E[E[X|\mathcal G]]$ is just another way of saying $E[X] = \int h(y,z) d\mu(y,z)$ where $h: \mathbb R^2 \to \mathbb R$ is some measurable function satisfying $E[X| \mathcal G] = h(Y,Z)$ and $\mu$ is the probability distribution of the joint $(Y,Z)$.
Why is Olympus Mons the largest volcano in the whole solar system? I will further expand on leonbloy's answer by emphasizing the role of change of variables for integrals, but this will be a self-contained answer. E[X]=E[E[X|Y]]. Can the hydrogen bond angle of water be changed via distillation? where $U$, $V$ and $W$ are any random variables. Concatenations of powers and their squares. The natural question arises, can I say something about my informed bet in average? Thanks for contributing an answer to Mathematics Stack Exchange! If it is more persuasive, we can imagine tossing the chosen coin $12$ times. How "Somewhere you are" authentication adds further security?
$E[X \mid Y]$ is itself a random variable $f(Y)$ where $$f(y) = E[X \mid Y = y) = \sum_x x\cdot Pr[X=x\mid Y=y].$$ Keeping this observation in mind, I still don't see why $U$ is "averaged out" when moving from the right hand side to the left side. Hot Network Questions Recognise a Digit from a Positional Encoding What reason would an extraterrestrial civilization have to adopt Earth's time measurement units? Later, we will see a deeper reason for this. If things go well, I will be able to look at it in a few days. Two-fifths of them favour heads, probability of head $0.8$. How "Somewhere you are" authentication adds further security? Can I become a tenure-track prof in one dept (biology) and teach in a different dept (math) with only one PhD? Assume at each of your exam, one moderator always takes a glance at your answer sheet and give you an assessment about how well you did. Unexpected goods delivery - is this a scam, and what kind of scam? Think of the inner term of the RHS : E[E[X|Y]] as integrating over a specific direction, then the outer expectation integrates over the remaining directions. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. p(A)=p(A|B)p(B)+p(A|~B)p(B) The σ-algebra $${\displaystyle {\mathcal {H}}}$$ could be replaced by a random variable $${\displaystyle Z}$$. So you are done. My question is: What is the intuitive meaning of this? What was "inertial" about the Inertial Upper Stage?
$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$? Why this weird definition of conditional expectation? Willingness to accept is different from willingness to pay - is there a name for this. well defined (as opposed to everywhere defined), and measurable functions that come from theorems that only guarantee uniqueness with respect to a.e. To learn more, see our tips on writing great answers. The new random variable likely has less variance in distribution if the moderator's observation is relatively accurate. Is conditional entropy ever taken to be a random variable? @Did, I'm confused as to your complaint about leonbloy's wording. To learn more, see our tips on writing great answers. For if we do that, in about $500$ cases we will get the first type of coin, and out of these $500$ we will get about $350$ heads, and so on. Sometimes you start with an informal argument for why you find the (meaningless) equation $E[X | Y = 2] = h(2)$ plausible and then you keep replacing steps in the informal argument into rigorous steps until you come up with a rigorous proof of the (not meaningless) equation $E[X|Y] = h(Y)$. This is yet another random variable, and it is E(X|Y). Proving that might require applying properties of conditional expectation several times until you arrive at the desired conclusion $E[X|Y] = h(Y)$. The tower property is more simply/generally expressed as $E[E[V | U]] = E[V]$. What was "inertial" about the Inertial Upper Stage? 3. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Swapping out our Syntax Highlighter, Intuitive explanation of the tower property of conditional expectation, non-measure theoretic proof of towering property of expectation. Let's say you successfully calculated $\mu$ and then also $\int_{-\infty}^{\infty} h(y) d\mu(y)$. In case I forget, please leave a reminder say three days from now. It only takes a minute to sign up. It doesn't become a constant since it is still conditioned on $U$. where the key missing word is "random"? Then average the answers, taking into consideration the proportions of the various types of coin. Also known as the law of total expectation. All the following formulas are to be understood in an almost sure sense. If lucky, you may be able to prove that $E[X|Y] = h(Y)$ holds almost everywhere. Hello highlight.js! Once the moderator tells you his assessment of your test, you obviously would have a different assessment of your test score. Now it only remains to rigorously prove that $\int_{-\infty}^{\infty} h(y) d\mu(y)$ is actually equal to $E(X)$ and you immediately see a little problem: the expectation along a particular slice such as $Y=2$ may have no meaning at all because $Y=2$ may be a null event. The theorem basically states that, if all of your models are correct, the extra information provided by the moderator would not change your average score, i.e. Do I have to start my story with my main characters? The expectation of the weight $E(X)$ would be my best guess of the weight of a unknown person: I'd bet for this value, if not given more data (my uninformed bet is constant).
For example, your math test scores can be modeled as a random variable distributed around a certain mean.
Proving $\mathbb{E}[(X - bY)^{2}]$ is minimized at $b = 1$. In this section we present a short list of important rules for manipulating and calculating conditional expectations. Measurability on subsets | tower property of conditional expectation. The example I came up with: take a die, now what is the average of values given they are odd: (1+3+5)/3 = 3, what is the average of values given they are even: (2+4+6)/3 = 4.
A general result along the same lines - called the tower property of con- ditional expectation - will be stated and proved below.
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