How do you calculate conditional expectation in Sigma algebra?
In general, we define E(X|Z) = E(X|σ(Z)), the conditional expected value given the sigma algebra generated by X, σ(X). We can define the condi- tional variance var(X|G) = E{(X − E(X|G))2|G}.
What does it mean to condition on a sigma algebra?
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection. of subsets of X satisfying the following conditions : (1) it includes X itself, (2) it is closed under complement, (3) it is closed under countable unions, and (4) it is closed under countable intersections.
How do you interpret conditional expectations?
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of “conditions” is known to occur.
What are the properties of expectation?
The following properties of expectation apply to discrete, continuous, and mixed random variables:
- Indicator function. The expectation of the indicator function is a probability: (5.56)
- Linearity. Expectation is a linear operator: (5.58)
- Nonnegative.
- Symmetry.
- Independence.
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).
Which of the following is are the properties of expectations?
If X and Y are the two variables, then the mathematical expectation of the sum of the two variables is equal to the sum of the mathematical expectation of X and the mathematical expectation of Y.
Is sigma algebra a field?
In fact field and sigma-field are algebra and sigma-algebra of Real Analysis in probability. The difference is in one condition. In Sigma-field you need being closed in respect of countable(finite and infinite countable) union but in field (without sigma) you only need being closed in respect of finite union.
Is a conditional probability random?
We show that, given a partition of events, conditional probability can be regarded as a random variable (probability conditional on a partition).
Is conditional expectation a function?
Conditional expectations such as E[X|Y = 2] or E[X|Y = 5] are numbers. If we consider E[X|Y = y], it is a number that depends on y. So it is a function of y.
What are the properties of expectation and variance?
Given a random variable, we often compute the expectation and variance, two important summary statistics. The expectation describes the average value and the variance describes the spread (amount of variability) around the expectation.
Can you multiply expectations?
Multiplying a random variable by any constant simply multiplies the expectation by the same constant, and adding a constant just shifts the expectation: E[kX+c] = k∙E[X]+c . For any event A, the conditional expectation of X given A is defined as E[X|A] = Σx x ∙ Pr(X=x | A) .