How to use conditional expectation in probability theory?

How to use conditional expectation in probability theory?

CONDITIONAL EXPECTATION: L2¡THEORY. Definition 1. Let (›,F,P) be a probability space and let G be a ¾¡algebra contained in F. For any real random variable X 2 L2(›,F,P), define E(X jG) to be the orthogonal projection of X. onto the closed subspace L2(›,G,P).

Which is the formula for a conditional PDF?

In the standard purely purely continuous case, there is a conditional pdf, which can be found from the formula p(y j x) = p(y;x) ∫ p(y;x)dy: 11

How are X and Y independent random variables?

Conversely, X and Y are independent random variables if for all x and y, their joint distribution function F(x, y) can be expressed as a prod- uct of a function of xalone and a function of yalone (which are the marginal distributions of andX Y, respec- tively).

What do you call a random variable that takes on infinite values?

A random variable that takes on a finite or countably infinite number of values (see page 4) is called a dis- crete random variable while one which takes on a noncountably infinite number of values is called a nondiscrete random variable.

How to write Epe with conditioning on X?

By conditioning on X, we can write EPE as and we see that it suffices to minimize EPE point-wise: the conditional expectation, also known as the regression function. The equation (2.11) is a consequence of the following little equality. For any two random variables Z1 and Z2, and any function g

What is the expected prediction error per below?

I am struggling to understand the derivation of the expected prediction error per below (ESL), especially on the derivation of 2.11 and 2.12 (conditioning, the step towards point-wise minimum). Any pointers or links much appreciated. Below I am reporting the excerpt from ESL pg. 18.

What is the expectation of a Cauchy random variable?

A Cauchy random variable takes a value in (−∞,∞) with the fol- lowing symmetric and bell-shaped density function. f(x) = 1 π[1+(x−µ)2] The expectation of Bernoulli random variable implies that since an indicator function of a random variable is a Bernoulli random variable, its expectation equals the probability.

Is the expectation of a random variable a linear operator?

In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].

Which is the expectation of proof of X?

Regarding your last question, the expectation can be either w.r.t. p(x, y) (the unconditional error) or w.r.t. p(y ∣ x) (the conditional error at each value X = x ). Happily, minimizing the conditional error at each value X = x also minimizes the unconditional error, so this is not a crucial distinction.

Is the conditional mean of X a random variable?

By definition, the conditional mean of Y on X is a random variable ψ with the following two properties: ψ lies in L2(Ω, FX, μ). E[ψ1A] = E[Y1A], for all A ∈ FX, which implies that E[ψg] = E[Yg], for all g ∈ L2(Ω, FX, μ), by standard argument use denseness of simple functions.

Which is the global minimizer of conditional expectations?

By the properties of conditional expectations we end up with ⇒ 0 ≤ [E(Y ∣ X) − h(x)]2 which holds with strict inequality if h(x) ≠ E(Y ∣ X). So E(Y ∣ X) is the global and unique minimizer.