*EM(Expectation Maximization ) жё…йЈЋеЏЇж‰ - еЌље®ўе› 2007-11-14 · X is a random variable, so is H(X). So we can talk about their expectation and variance. Of particular interest are g(X) = E[YjX] and h(X) = Var(YjX) There are two important theorems about these quantities Theorem. Iterated Expectation E[E[XjY]] = E[X] Conditional Expectation 3*

Conditional Expectation Functions UCSB Department of. 2009-11-11 · Probability 2 - Notes 5 Conditional expectations E(XjY) as random variables Conditional expectations were discussed in lectures (see also the second part of Notes 3). The, 2009-9-24 · Expectation of Random Variables September 17 and 22, 2009 1 Discrete Random Variables Let x 1;x 2; x n be observation, the empirical mean, x = 1 n (x 1 + x Using the linearity of expectation ES= E[X 1 + X 2 + X n] = p+ p+ + p= np: 1.1 Discrete Calculus Let hbe a function on whose domain and range are integers. The (positive) di erence operator.

2005-2-3 · Intuition behind the Law of Iterated Expectations • Simple version of the law of iterated expectations (from Wooldridge’s Econometric Analysis of Cross Section and Panel Data, p. 29): 2009-11-11 · Probability 2 - Notes 5 Conditional expectations E(XjY) as random variables Conditional expectations were discussed in lectures (see also the second part of Notes 3). The

2006-11-20 · x· PDF R(x). Let’s work through an example. Let R be the number that comes up on a fair, six-sided die. 2 Course Notes, Week 13: Expectation & Variance The proof of Theorem 1.2, like many of the elementary proofs about expectation in these notes, follows by judicious regrouping of terms in the deﬁning sum (1): 2010-11-18 · STA 205 Conditional Expectation R L Wolpert λa(dx) = Y(x)dx with pdf Y and a singular part λs(dx) (the sum of the singular-continuous and discrete components). When λ ≪ µ (so λa = λ and λs = 0) the Radon-Nikodym derivative is often denoted Y = dλ dµ or λ(dω) µ(dω), and extends the idea of “density” from densities with respect to Lebesgue

2019-11-13 · ExpectationThe expectation is the expected value of X, written as E(X) or sometimes as μ.The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the 'mean'.To calculate the expectation we can use the following formula:E(X) = ∑ xP(X = x)It may look complicated, but in fact is quite easy to use.You multiply each value of x 2018-1-25 · Conditional Expectation Functions Econometrics II Douglas G. Steigerwald UC Santa Barbara D. Steigerwald (UCSB) Expectation Functions 1 / 32. is a function of the random variable x only I to calculate its expectation, integrate with respect …

2014-10-28 · the utility gained from each additional dollar decreases with the wealth. Under diminishing marginalutility,u(2X) growsslowerthan2X,andthereforetheexpectedvalueEu(2X) canbe ﬁnite. Forexample,withalog utilityfunction,onecanshowthatElog(2X) = 2log(2) ˇ1:39.1 In this example, suppose that the cost of participation in the game is c. 2008-4-25 · Expectation of geometric distribution What is the probability that X is nite? 1 k=1fX(k) = 1k=1(1 p) k 1p = p 1 j=0(1 p) j = p 1 1 (1 p) = 1 Can now compute E(X): E(X) = 1 geometric distribution! Bottom line: the algorithm is extremely fast and almost certainly gives the right results. 9

2019-9-29 · 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. If the random variable can take on only a finite number of values, the “conditions” are that 2014-8-18 · 2. Conditional expectation: the expectation of a random variable X, condi-tional on the value taken by another random variable Y. If the value of Y aﬀects the value of X (i.e. X and Y are dependent), the conditional expectation of X given the value of Y will be diﬀerent from the overall expectation of X. 3.

2012-10-10 · EM 的意思是 “Expectation-Maximazation”，在这个聚类问题里面，我们是先随便猜一下这两个正态分布的参数：如核心在什么地方，方差是多少。然后计算出每个数据点更可能属于第一个还是第二个正态分布圈，这个是属于 Expectation 一步。 2019-11-13 · ExpectationThe expectation is the expected value of X, written as E(X) or sometimes as μ.The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the 'mean'.To calculate the expectation we can use the following formula:E(X) = ∑ xP(X = x)It may look complicated, but in fact is quite easy to use.You multiply each value of x

2005-2-3 · Intuition behind the Law of Iterated Expectations • Simple version of the law of iterated expectations (from Wooldridge’s Econometric Analysis of Cross Section and Panel Data, p. 29): 2014-8-18 · 2. Conditional expectation: the expectation of a random variable X, condi-tional on the value taken by another random variable Y. If the value of Y aﬀects the value of X (i.e. X and Y are dependent), the conditional expectation of X given the value of Y will be diﬀerent from the overall expectation of X. 3.

Intuition behind the Law of Iterated Expectations. 2019-8-15 · Lecture 6 Gamma distribution, 2-distribution, Student t-distribution, Fisher F -distribution. Gamma distribution. Let us take two parameters > 0 and > 0. Gamma function ( ) is deﬁned by ( ) = x −1e−xdx. 0 If we divide both sides by ( ) we get 1 1 = x −1e −xdx = y e ydy 0 0, 2014-8-18 · 2. Conditional expectation: the expectation of a random variable X, condi-tional on the value taken by another random variable Y. If the value of Y aﬀects the value of X (i.e. X and Y are dependent), the conditional expectation of X given the value of Y will be diﬀerent from the overall expectation of X. 3..

1. The Uniform Distribution Imperial College London. 2018-8-24 · Transformations and Expectations of random variables X˘F X(x): a random variable Xdistributed with CDF F X. Any function Y = g(X) is also a random variable. If both X, and Y are continuous random variables, can we nd a simple way to characterize, 2004-5-5 · 12.3: Expected Value and Variance If X is a random variable with corresponding probability density function f(x), then we deﬁne the expected value of X to be E(X) := Z ∞ −∞ xf(x)dx We deﬁne the variance of X to be Var(X) := Z ∞ −∞ [x − E(X)]2f(x)dx 1 Alternate formula for the variance As with the variance of a discrete random.

1 Expectation of a discrete random variable. 2018-8-15 · X to emphasize that the expectation is taken with respect to a particular random variable X. For a continuous random variable, the expectation is sometimes written as, E[g(X)] = Z x −∞ g(x) dF(x). where F(x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral. https://ru.wikipedia.org/wiki/EM-%D0%B0%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC 2018-1-25 · Conditional Expectation Functions Econometrics II Douglas G. Steigerwald UC Santa Barbara D. Steigerwald (UCSB) Expectation Functions 1 / 32. is a function of the random variable x only I to calculate its expectation, integrate with respect ….

2016-11-3 · The expectation operator maps a function of a random variable or of several random variables to an average weighted by the corresponding pmf or pdf. De nition 2.1 (Expectation for discrete random variables). Let X be a discrete random variable with range R. The expected value of a function g(X), g: R !R, of Xis E(g(X)) = X x2R g(x)p X(x): (1) 2018-8-15 · X to emphasize that the expectation is taken with respect to a particular random variable X. For a continuous random variable, the expectation is sometimes written as, E[g(X)] = Z x −∞ g(x) dF(x). where F(x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral.

2005-2-3 · Intuition behind the Law of Iterated Expectations • Simple version of the law of iterated expectations (from Wooldridge’s Econometric Analysis of Cross Section and Panel Data, p. 29): 2014-10-28 · the utility gained from each additional dollar decreases with the wealth. Under diminishing marginalutility,u(2X) growsslowerthan2X,andthereforetheexpectedvalueEu(2X) canbe ﬁnite. Forexample,withalog utilityfunction,onecanshowthatElog(2X) = 2log(2) ˇ1:39.1 In this example, suppose that the cost of participation in the game is c.

2011-11-18 · Circular Uniform PDF. Ben throws a dart at a circular target of radius . We assume that he always hits the target, and that all points of impact are equally likely, so that the joint PDF of the random variables and is uniform – What is the marginal PDF f X,Y x, y f Y y 2019-10-17 · Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange

2019-8-15 · Lecture 6 Gamma distribution, 2-distribution, Student t-distribution, Fisher F -distribution. Gamma distribution. Let us take two parameters > 0 and > 0. Gamma function ( ) is deﬁned by ( ) = x −1e−xdx. 0 If we divide both sides by ( ) we get 1 1 = x −1e −xdx = y e ydy 0 0 2017-4-4 · Law of Iterated Expectations Guillem Riambau. YSS211. Econometrics, Yale-NUS. Spring 2016. The Law of Iterated Expectations states that: (1) E(X) = E(E(XjY)) This document tries to give some intuition to the L.I.E. Sometimes you may see it written as E(X) = E y(E x(XjY)). At the end of the document it is explained why (note, both mean exactly

2008-1-24 · The partition theorem says that if Bn is a partition of the sample space then E[X] = X n E[XjBn]P(Bn) Now suppose that X and Y are discrete RV’s. If y is in the range of Y then Y = y is a event with nonzero probability, so we can use it as the B in the above. 2009-11-11 · Probability 2 - Notes 5 Conditional expectations E(XjY) as random variables Conditional expectations were discussed in lectures (see also the second part of Notes 3). The

2014-10-28 · the utility gained from each additional dollar decreases with the wealth. Under diminishing marginalutility,u(2X) growsslowerthan2X,andthereforetheexpectedvalueEu(2X) canbe ﬁnite. Forexample,withalog utilityfunction,onecanshowthatElog(2X) = 2log(2) ˇ1:39.1 In this example, suppose that the cost of participation in the game is c. 2013-9-27 · Random Variables and Expectation A random variable arises when we assign a numeric value to each elementary event. For example, if each elementary event is the result of a series of three tosses of a fair coin, then X = “the number of Heads” is a random variable.

2016-11-3 · The expectation operator maps a function of a random variable or of several random variables to an average weighted by the corresponding pmf or pdf. De nition 2.1 (Expectation for discrete random variables). Let X be a discrete random variable with range R. The expected value of a function g(X), g: R !R, of Xis E(g(X)) = X x2R g(x)p X(x): (1) 2011-1-31 · X(a)≡E(X) Motivates deﬁnition of expectation, suggests that expectation characterizes long term behavior of sample averages of random processes To make precise, need to consider limits of random variables — different from usual deﬁnition of limits of sequences of real numbers First: develop expectation and its properties in more detail

2018-8-24 · Transformations and Expectations of random variables X˘F X(x): a random variable Xdistributed with CDF F X. Any function Y = g(X) is also a random variable. If both X, and Y are continuous random variables, can we nd a simple way to characterize 2018-8-24 · Transformations and Expectations of random variables X˘F X(x): a random variable Xdistributed with CDF F X. Any function Y = g(X) is also a random variable. If both X, and Y are continuous random variables, can we nd a simple way to characterize

2019-3-28 · Minimize MSE=E{[Y−h(X)]}. For every x, the inner expectation is minimized by setting h(x) equal to the constant E(Y X=x), according to the preceding trivial case. Unfortunately, this optimal prediction scheme depends on knowing the joint distribution of Y and X 2018-8-15 · X to emphasize that the expectation is taken with respect to a particular random variable X. For a continuous random variable, the expectation is sometimes written as, E[g(X)] = Z x −∞ g(x) dF(x). where F(x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral.

properties of variance University of Washington. 2017-4-4 · Law of Iterated Expectations Guillem Riambau. YSS211. Econometrics, Yale-NUS. Spring 2016. The Law of Iterated Expectations states that: (1) E(X) = E(E(XjY)) This document tries to give some intuition to the L.I.E. Sometimes you may see it written as E(X) = E y(E x(XjY)). At the end of the document it is explained why (note, both mean exactly, 2019-11-13 · ExpectationThe expectation is the expected value of X, written as E(X) or sometimes as μ.The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the 'mean'.To calculate the expectation we can use the following formula:E(X) = ∑ xP(X = x)It may look complicated, but in fact is quite easy to use.You multiply each value of x.

The EM(ExpectationвЂ“Maximization) Algorithm . 2004-4-16 · Key Point The Uniform random variable X whose density function f(x)isdeﬁned by f(x)= 1 b−a,a≤ x ≤ b 0 otherwise has expectation and variance given by the formulae E(X)= b+a 2 and V(X)= (b−a)212 Example The current (in mA) measured in a piece of copper wire is known to follow a uniform distribution over the interval [0,25].Write down the formula for, 2009-9-24 · Expectation of Random Variables September 17 and 22, 2009 1 Discrete Random Variables Let x 1;x 2; x n be observation, the empirical mean, x = 1 n (x 1 + x Using the linearity of expectation ES= E[X 1 + X 2 + X n] = p+ p+ + p= np: 1.1 Discrete Calculus Let hbe a function on whose domain and range are integers. The (positive) di erence operator.

2006-11-20 · x· PDF R(x). Let’s work through an example. Let R be the number that comes up on a fair, six-sided die. 2 Course Notes, Week 13: Expectation & Variance The proof of Theorem 1.2, like many of the elementary proofs about expectation in these notes, follows by judicious regrouping of terms in the deﬁning sum (1): 2007-11-14 · X is a random variable, so is H(X). So we can talk about their expectation and variance. Of particular interest are g(X) = E[YjX] and h(X) = Var(YjX) There are two important theorems about these quantities Theorem. Iterated Expectation E[E[XjY]] = E[X] Conditional Expectation 3

2011-1-31 · X(a)≡E(X) Motivates deﬁnition of expectation, suggests that expectation characterizes long term behavior of sample averages of random processes To make precise, need to consider limits of random variables — different from usual deﬁnition of limits of sequences of real numbers First: develop expectation and its properties in more detail 2010-11-18 · STA 205 Conditional Expectation R L Wolpert λa(dx) = Y(x)dx with pdf Y and a singular part λs(dx) (the sum of the singular-continuous and discrete components). When λ ≪ µ (so λa = λ and λs = 0) the Radon-Nikodym derivative is often denoted Y = dλ dµ or λ(dω) µ(dω), and extends the idea of “density” from densities with respect to Lebesgue

2008-1-24 · The partition theorem says that if Bn is a partition of the sample space then E[X] = X n E[XjBn]P(Bn) Now suppose that X and Y are discrete RV’s. If y is in the range of Y then Y = y is a event with nonzero probability, so we can use it as the B in the above. 2012-10-10 · EM 的意思是 “Expectation-Maximazation”，在这个聚类问题里面，我们是先随便猜一下这两个正态分布的参数：如核心在什么地方，方差是多少。然后计算出每个数据点更可能属于第一个还是第二个正态分布圈，这个是属于 Expectation 一步。

2013-9-27 · Random Variables and Expectation A random variable arises when we assign a numeric value to each elementary event. For example, if each elementary event is the result of a series of three tosses of a fair coin, then X = “the number of Heads” is a random variable. 2019-6-21 · Conditional expectation. by Marco Taboga, PhD. The conditional expectation (or conditional mean, or conditional expected value) of a random variable is the expected value of the random variable itself, computed with respect to its conditional probability distribution.. As in the case of the expected value, a completely rigorous definition of conditional expected value requires a complicated

2018-8-15 · X to emphasize that the expectation is taken with respect to a particular random variable X. For a continuous random variable, the expectation is sometimes written as, E[g(X)] = Z x −∞ g(x) dF(x). where F(x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral. 2018-1-25 · Conditional Expectation Functions Econometrics II Douglas G. Steigerwald UC Santa Barbara D. Steigerwald (UCSB) Expectation Functions 1 / 32. is a function of the random variable x only I to calculate its expectation, integrate with respect …

2019-6-21 · Conditional expectation. by Marco Taboga, PhD. The conditional expectation (or conditional mean, or conditional expected value) of a random variable is the expected value of the random variable itself, computed with respect to its conditional probability distribution.. As in the case of the expected value, a completely rigorous definition of conditional expected value requires a complicated 2011-5-3 · CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time.

2005-2-3 · Intuition behind the Law of Iterated Expectations • Simple version of the law of iterated expectations (from Wooldridge’s Econometric Analysis of Cross Section and Panel Data, p. 29): 2019-11-13 · ExpectationThe expectation is the expected value of X, written as E(X) or sometimes as μ.The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the 'mean'.To calculate the expectation we can use the following formula:E(X) = ∑ xP(X = x)It may look complicated, but in fact is quite easy to use.You multiply each value of x

2019-8-15 · Lecture 6 Gamma distribution, 2-distribution, Student t-distribution, Fisher F -distribution. Gamma distribution. Let us take two parameters > 0 and > 0. Gamma function ( ) is deﬁned by ( ) = x −1e−xdx. 0 If we divide both sides by ( ) we get 1 1 = x −1e −xdx = y e ydy 0 0 2004-4-16 · Key Point The Uniform random variable X whose density function f(x)isdeﬁned by f(x)= 1 b−a,a≤ x ≤ b 0 otherwise has expectation and variance given by the formulae E(X)= b+a 2 and V(X)= (b−a)212 Example The current (in mA) measured in a piece of copper wire is known to follow a uniform distribution over the interval [0,25].Write down the formula for

1. The Uniform Distribution Imperial College London. 2018-8-24 · Transformations and Expectations of random variables X˘F X(x): a random variable Xdistributed with CDF F X. Any function Y = g(X) is also a random variable. If both X, and Y are continuous random variables, can we nd a simple way to characterize, 2012-8-13 · G. EXPECTATION RULES AND DEFINITIONS. a, b are any given constants. X, Y are random variables. The following apply. [NOTE: we’ll use a few of these now and others.

Conditional Expectation Duke University. 2011-1-31 · X(a)≡E(X) Motivates deﬁnition of expectation, suggests that expectation characterizes long term behavior of sample averages of random processes To make precise, need to consider limits of random variables — different from usual deﬁnition of limits of sequences of real numbers First: develop expectation and its properties in more detail, 2005-2-3 · Intuition behind the Law of Iterated Expectations • Simple version of the law of iterated expectations (from Wooldridge’s Econometric Analysis of Cross Section and Panel Data, p. 29):.

Conditional Expectation Duke University. 2019-11-6 · Note: X given Y = y is defined in the same way (just switch the variables). The formula might look a little daunting, but it’s actually pretty simple to work. What it is telling you to do is find the proportions of the “conditional” part (all the values where X = …, 2019-9-5 · 2.1 Expectation Equals Arithmetic Mean Expectation is defined as $1^{st}$ raw moment: Expectation is the arithmetic mean of any random variable coming from any probability distribution，这个不用怀疑，可以参见这篇 Why is expectation the same as the。.

Conditional Expectation Functions UCSB Department of. 2012-9-17 · The mean or expected value of X is defined by E(X) = sum x k p(x k). Expectation of a function of a random variable Let X be a random variable assuming the values x 1, x 2, x if X is continuous with pdf f(x) ()iXisdiscretewith p mf p(x) ()() efxdx ex MtEe tx x tx tX X The reason M X (t) https://en.wikipedia.org/wiki/Expectation_(statistics) 2017-4-4 · Law of Iterated Expectations Guillem Riambau. YSS211. Econometrics, Yale-NUS. Spring 2016. The Law of Iterated Expectations states that: (1) E(X) = E(E(XjY)) This document tries to give some intuition to the L.I.E. Sometimes you may see it written as E(X) = E y(E x(XjY)). At the end of the document it is explained why (note, both mean exactly.

2006-11-20 · x· PDF R(x). Let’s work through an example. Let R be the number that comes up on a fair, six-sided die. 2 Course Notes, Week 13: Expectation & Variance The proof of Theorem 1.2, like many of the elementary proofs about expectation in these notes, follows by judicious regrouping of terms in the deﬁning sum (1): 2019-8-15 · 18.05 class 6, Expectation and Variance for Continuous Random Variables 2 So f(x)dxrepresents the probability that Xis in an in nitesimal range of width dxaround x. Thus we can interpret the formula for E(X) as a weighted integral of the values xof X, where the weights are the probabilities f(x)dx.

2010-11-18 · STA 205 Conditional Expectation R L Wolpert λa(dx) = Y(x)dx with pdf Y and a singular part λs(dx) (the sum of the singular-continuous and discrete components). When λ ≪ µ (so λa = λ and λs = 0) the Radon-Nikodym derivative is often denoted Y = dλ dµ or λ(dω) µ(dω), and extends the idea of “density” from densities with respect to Lebesgue 2019-10-17 · Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange

2011-1-31 · X(a)≡E(X) Motivates deﬁnition of expectation, suggests that expectation characterizes long term behavior of sample averages of random processes To make precise, need to consider limits of random variables — different from usual deﬁnition of limits of sequences of real numbers First: develop expectation and its properties in more detail 2012-8-13 · G. EXPECTATION RULES AND DEFINITIONS. a, b are any given constants. X, Y are random variables. The following apply. [NOTE: we’ll use a few of these now and others

2005-2-3 · Intuition behind the Law of Iterated Expectations • Simple version of the law of iterated expectations (from Wooldridge’s Econometric Analysis of Cross Section and Panel Data, p. 29): 2019-3-28 · Minimize MSE=E{[Y−h(X)]}. For every x, the inner expectation is minimized by setting h(x) equal to the constant E(Y X=x), according to the preceding trivial case. Unfortunately, this optimal prediction scheme depends on knowing the joint distribution of Y and X

2012-8-13 · G. EXPECTATION RULES AND DEFINITIONS. a, b are any given constants. X, Y are random variables. The following apply. [NOTE: we’ll use a few of these now and others 2008-1-24 · The partition theorem says that if Bn is a partition of the sample space then E[X] = X n E[XjBn]P(Bn) Now suppose that X and Y are discrete RV’s. If y is in the range of Y then Y = y is a event with nonzero probability, so we can use it as the B in the above.

2019-9-5 · 2.1 Expectation Equals Arithmetic Mean Expectation is defined as $1^{st}$ raw moment: Expectation is the arithmetic mean of any random variable coming from any probability distribution，这个不用怀疑，可以参见这篇 Why is expectation the same as the。 2008-4-25 · Expectation of geometric distribution What is the probability that X is nite? 1 k=1fX(k) = 1k=1(1 p) k 1p = p 1 j=0(1 p) j = p 1 1 (1 p) = 1 Can now compute E(X): E(X) = 1 geometric distribution! Bottom line: the algorithm is extremely fast and almost certainly gives the right results. 9

2019-11-5 · In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values x i and corresponding probabilities p i.Now consider a weightless rod on which are placed weights, at locations x i along the rod and having masses p i (whose sum is one). The point at which the rod balances is E[X]. 2004-5-5 · 12.3: Expected Value and Variance If X is a random variable with corresponding probability density function f(x), then we deﬁne the expected value of X to be E(X) := Z ∞ −∞ xf(x)dx We deﬁne the variance of X to be Var(X) := Z ∞ −∞ [x − E(X)]2f(x)dx 1 Alternate formula for the variance As with the variance of a discrete random

2019-10-17 · Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange 2009-11-11 · Probability 2 - Notes 5 Conditional expectations E(XjY) as random variables Conditional expectations were discussed in lectures (see also the second part of Notes 3). The

2007-11-14 · X is a random variable, so is H(X). So we can talk about their expectation and variance. Of particular interest are g(X) = E[YjX] and h(X) = Var(YjX) There are two important theorems about these quantities Theorem. Iterated Expectation E[E[XjY]] = E[X] Conditional Expectation 3 2011-1-31 · X(a)≡E(X) Motivates deﬁnition of expectation, suggests that expectation characterizes long term behavior of sample averages of random processes To make precise, need to consider limits of random variables — different from usual deﬁnition of limits of sequences of real numbers First: develop expectation and its properties in more detail