The properties above for conditional expected value, of course, have special cases for conditional probability. is a a . In this setting above suppose that \( \E(|X|) \lt \infty \). such that \[ \P(A \mid \mathscr G) = \E(\bs 1_A \mid \mathscr G) \]. It remains at 1 another exponentially distributed random time, T2, which is independent of T1, and at time T1 + T2 it jumps from 1 to 2, and so on. Then take expected values through the inequality. If

Hopefully this idea will become clearer during our study. the random vector In turn, this property leads to a formula for the mean square error when \( \E(X \mid \mathscr G) \) is thought of as a predictor of \( X \). Moreover, the random variables that are measurable with respect to \( \mathscr G \) are precisely the variables that are constant on \( A_i \) for each \( i \in I \). follows: The linearity property of the expected In our elementary treatment of conditional expected value, we showed that the conditional expected value of a real-valued random variable \( X \) given a general random variable \( Y \) is the best predictor of \( X \), in the least squares sense, among all real-valued functions of \( Y \). Our next properties are fundamental: every version of expected value must satisfy the linearity properties. The following theorem gives the conditional version of the axioms of probability. Thus, by the previous property

The solution is the conditional expectation H(X) = E(Y|X). entries of a Then. By matrix of constants, If \( U \) is measurable with respect to \( \mathscr G \) and \( \E(|U X|) \lt \infty \) then \( \E[U \E(X \mid \mathscr G)] = \E(U X) \). is easily proved by applying the linearity properties above to each entry of constants:Let More precisely from this point of view, the objects of our study are not individual random variables but rather equivalence classes of random variables under this equivalence relation. Then \(\newcommand{\sd}{\text{sd}}\)

The \( \sigma \)-algebra \( \mathscr G \) is said to be countably generated.



Thus, it should come as not surprise that if \( \mathscr G \) is a sub \( \sigma \)-algebra of \( \mathscr F \), then \( \E(X \mid \mathscr G) \) is the best predictor of \( X \), in the least squares sense, among all real-valued random variables that are measurable with respect to \( \mathscr G) \).

is a constant, is a An important problem of probability theory is to predict the value of a future observation Y given knowledge of a related observation X (or, more generally, given several related observations X1, X2,…).

Two properties of the Poisson process that make it attractive to deal with theoretically are: (i) The times between emission of particles are independent and exponentially distributed with expected value 1/μ. be a random variable with expected Suppose that \(X\) is a random variable with \( \E(|X|) \lt \infty \). next the notion of equivalence plays a fundamental role in this section. \begin{align*} matrix whose entries are random variables. Let Suppose again that \( \mathscr G \) is a sub \( \sigma \)-algebra of \( \mathscr F \). As usual, let 1(A) denote the indicator random variable of A. W e define ℙ(A||X) = (1(A)||X) The properties above for conditional expected value, of course, have special cases for conditional … If \( V \) is measurable with respect to \( \mathscr G \) then \( V \) is like a constant in terms of the conditional expected value given \( \mathscr G \).