MullOverThings

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How does conditional expectation work?

How does conditional expectation work?

The conditional expectation (also called the conditional mean or conditional expected value) is simply the mean, calculated after a set of prior conditions has happened.

How do you find conditional expectation continuously?

Exercise: Prove E[Y g(X)] = E[E[Y |X]g(X)] if X and Y are jointly continuous random variables. The conditional expectation E[Y |X] can be viewed as an estimator of Y given X. Y − E(Y |X) is then the estimation error for this estimator.

Is conditional expectation random?

Conditional expectations such as E[X|Y = 2] or E[X|Y = 5] are numbers. ω → E[X|Y = y] 2 Page 3 So this is a random variable. It is usually written as E[X|Y ].

How do you find conditional variance?

Similar to the conditional expectation, we can define the conditional variance of X, Var(X|Y=y), which is the variance of X in the conditional space where we know Y=y. If we let μX|Y(y)=E[X|Y=y], then Var(X|Y=y)=E[(X−μX|Y(y))2|Y=y]=∑xi∈RX(xi−μX|Y(y))2PX|Y(xi)=E[X2|Y=y]−μX|Y(y)2.

What is the formula for conditional probability?

The formula for conditional probability is derived from the probability multiplication rule, P(A and B) = P(A)*P(B|A). You may also see this rule as P(A∪B). The Union symbol (∪) means “and”, as in event A happening and event B happening.

What is expectation function?

Using expectation, we can define the moments and other special functions of a random variable. Definition 2 Let X and Y be random variables with their expectations µX = E(X) and µY = E(Y ), and k be a positive integer. 1. The kth moment of X is defined as E(Xk). If k = 1, it equals the expectation.

How do you calculate conditional distribution?

First, to find the conditional distribution of X given a value of Y, we can think of fixing a row in Table 1 and dividing the values of the joint pmf in that row by the marginal pmf of Y for the corresponding value. For example, to find pX|Y(x|1), we divide each entry in the Y=1 row by pY(1)=1/2.

Is conditional expectation linear?

The next lemma shows that conditional expectation is linear. Lemma 19 (Linearity). If E(X), E(Y ), and E(X + Y ) all exist, then E(X|C) + E(Y |C) is a version of E(X + Y |C).

What is the conditional mean of Y?

In contrast, the conditional mean of yt is the expected value of yt given a conditioning set of variables, Ωt. A conditional mean model specifies a functional form for E ( y t | Ω t ) . .

What is the difference between conditional and unconditional variance?

While the unconditional variance is just the standard measure of the variance, the conditional variance represents the measure of the uncertainty about a variable given a model and an information set .

How do you find the conditional variance example?

Conditional Variance as a Random Variable: As with E(Y|X), we can consider Var(Y|X) as a random variable. For example, if Y = height and X = sex for persons in a certain population, then Var(height | sex) is the variable which assigns to each person in the population the variance of height for that person’s sex.

What is an example of conditional probability?

Conditional probability: p(A|B) is the probability of event A occurring, given that event B occurs. Example: given that you drew a red card, what’s the probability that it’s a four (p(four|red))=2/26=1/13. So out of the 26 red cards (given a red card), there are two fours so 2/26=1/13.