# What is Conditional expectation: Definition and 60 Discussions

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 the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.
Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted

E
(
X

Y
)

{\displaystyle E(X\mid Y)}
analogously to conditional probability. The function form is either denoted

E
(
X

Y
=
y
)

{\displaystyle E(X\mid Y=y)}
or a separate function symbol such as

f
(
y
)

{\displaystyle f(y)}
is introduced with the meaning

E
(
X

Y
)
=
f
(
Y
)

{\displaystyle E(X\mid Y)=f(Y)}
.

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1. ### I Finding expectation of the function of a sum

My answer: Is the above answer correct?
2. ### Conditional expectation given ##\mathcal{F}_m##

Are these above answers correct?
3. ### Stochastic Calculus: Conditional Expectation

Are my following answers correct?

10. ### A Conditional expectation and covariance of function of RVs

Hey all, I have been doing some math lately where I need to find the conditional expectation of a function of random variables. I also at some point need to find a derivative with respect to the variable that has been conditioned. I am not sure of my work and would appreciate it if you guys can...
11. ### A Parameterizing conditional expectations (Gaussian case)

Consider three jointly normally distributed random variables X,Y and Z. I know that in the Gaussian case E[Z | X,=x Y=y]=xßZX;Y +yßZY;X where ßZX;Y notes the regression coefficient of Z on X conditional on Y (and ßZY;Xis analogously defined). Is the following derivation correct? E[Z| X>x...
12. ### Conditional Expectation of Multiple Independent Random Varia

Homework Statement Given X,Y,Z are 3 N(1,1) random variables, (1) Find E[ XY | Y + Z = 1] Homework EquationsThe Attempt at a Solution I'm honestly completely lost in statistics... I didn't quite grasp the intuitive aspect of expectation because my professor lives in the numbers side and...
13. ### Optimizing Conditional Expectation

Hi all, Let X be a random EDIT variable with (infinite) sample space S. Are there some results dealing with how to maximize E(X|s ) (conditional expectation of X given s ) for s in S ? Thanks.
14. ### Probability Conditional Expectation

Suppose X and Y are independent Poisson random variables with respective parameters λ and 2λ. Find E[Y − X|X + Y = 10]3: I had my Applied Probability Midterm today and this question was on it. The class is only 14 people and no one I talked to did it correctly. The prof sent out an e-mail saying...
15. ### Conditional expectation on an indicator

Homework Statement Let X and Y be independent Bernoulli RV's with parameter p. Find, \mathbb{E}[X\vert 1_{\{X+Y=0\}}] and \mathbb{E}[Y\vert 1_{\{X+Y=0\}}] Homework EquationsThe Attempt at a Solution I'm trying to show that, \mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] = 0 by, \begin{align*}...
16. ### Calculate conditional expectation of exponential variables

Homework Statement Let X and Y be independent exponential random variables with parameters a and b. Calculate E(X|X+Y). Homework EquationsThe Attempt at a Solution I'm pretty sure I have it, just want to make sure. Joint density for X and Y is abe^(-ax)e^(-by) for x,y>0. Let Z=X and W=X+Y so...
17. ### Conditional Expectation problem

1. I have a problem that I cannot figure out how to solve. I want to find the following: E(X|X<Y) where X follows exp(a) and Y follows exp(b) (exp is for exponential distribution). Any ideas on how to solve it? [b]I got E(X|X<Y) = \int_{-∞}^{∞} E(X|X<y)f_{y}(y)dy = \int_{-∞}^{∞}...
18. ### For which joint distributions is a conditional expectation an additive

I know that, for a random vector (X,Y,Z) jointly normally distributed, the conditional expectation E[X|Y=y,Z=z] is an additive function of y and z For what other distributions is this true?
19. ### MHB Find conditional expectation

Consider a family of densitites $f(x,\theta)=\frac{exp(-{\sqrt{x}})}{{\theta}}$. Let $X_{1}$ be a single observation from this family. I have shown that ${\sqrt{X_{1}}}/2$ is an unbiased estimator. Now consider $n$ observations $X_{1},..X_{n}$. I have shown that...
20. ### MHB Conditional expectation of joint PMF

(please refer to attached image) The question appears to be simple enough, but i have two queries A) does E[X1 X2] mean the same as E[X1 | X2] B) If not/so, how exactly do I go about computing this. I've seen a few formulas in my lectures notes for computing conditional expectations for...
21. ### MHB Conditional expectation proof question

Here is a proof question: For two random variables X and Y, we can define E(X|Y) to be the function of Y that satisfies E(Xg(X)) = E(E(X|Y)g(Y)) for any function g. Using this definition show that E(X1 + X2|Y) = E(X1|Y) + E(X2|Y) So what I did was I plugged into X = X1 + X2 E(E(X1 +...
22. ### A simple conditional expectation question

Let v be a random variable distributed according to F(.). Let X be a set containing the objects x1 and x2. Suppose E(v|x1) = b AND E(v|x2) = b (The expected value of v conditional on x1 is b, etc) where b is some constant. Does it follow that E(v|x1,x2) = b? If so, why...
23. ### Conditional expectation in statistics

Hi, I am trying to show that if the E[W|X]=0 then the Cov (W,X)=0. Using the def of variance, and given that E[W] is zero, I get that Cov is equal to: E[WX]-E[W * E(X)] using conditional expectation: E [E(WX|X)] -E[x]E[W]= E[X E[W|X]]-E[X]E[E(W|X)]=0 I am not sure if...
24. ### Conditional Expectation of a random variable

My professor made a rather concise statement in class, which sums to this: E(Y|X=xi) = constant. E(Y|X )= variable. Could anyone help me understand how the expectation is calculated for the second case? I understand that for different values of xi, we'll have different values for the...
25. ### Calculating Conditional Expectation of X,Y,Z: Triangle Inequality

1. Let T = (X,Y,Z) be a Gaussian for which X,Y,Z for which X, Y, Z are standard normals, such that E[XY] = E[YZ] = E[XZ] = 1/2. A) Calculate the characteristics function Φ_T(u,v,w) of T. B) Calculate the density of T. 2. Let X and Y be N(0,1) (standard normals), not necessarily...
26. ### Conditional expectation of a product of two independent random variables

Suppose that α and β are independently distributed random variables, with means; μ_α, μ_b and variances; δ_α^2, δ_β^2, respectively. Further, let c=αβ+e, where e is independently distributed from α and β with mean 0 and variance δ_e^2. Does it hold that E(αβ | c) = E(α|c)...
27. ### Conditional Expectation

1. Let the joint pdf be f(x,y) = 2 ; 0<x<y<1 ; 0<y<1 Find E(Y|x) and E(X|y)Homework Equations E(Y|x) = \int Y*f(y|x)dy f(y|x) = f(x,y) / f(x) The Attempt at a Solution f(x) = \int 2dy from 0 to y = 2y f(y|x) = f(x,y)/f(x) = 1/2y E(Y|x) = \int Y/2Y dy from x to 1 = \int 1/2 dy from x to 1 =...
28. ### MHB Proving conditional expectation

Hi guys, assume we have an equality involving 2 random variables U and X such that E(U|X) = E(U)=0, now I was told that this assumption implies that E(U^2|X) = E(U^2). However I'm not sure on how to prove this, if anyone could show me that'd be great!
29. ### Probability - Conditional Expectation

My professor explained this concept absolutely horribly and I have no idea how to do these problems. Let A and B be independent Poisson random variables with parameters α and β, respectively. Find the conditional expectation of A given A + B = c. (Hint: For discrete random variables, there...
30. ### Probability: Conditional expectation

Homework Statement What is the expected number of flips of a biased coin with probability of heads 'p', until two consecutive flips are heads?Homework Equations The Attempt at a Solution Let T_1 = first flip is tails, H_1 = first flip is heads. and T_2, H_2 for second flip. \mathbb{E}[X] =...
31. ### Conditional Expectation Problem

Homework Statement Suppose that X and Y have a continuous joint distribution with joint pdf given by f (x, y) = { x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 0 otherwise. Suppose that a person can pay a cost c for the opportunity of observing the value of X before...
32. ### Conditional expectation, Lebesgue measure

Homework Statement Let Ω = [0,1] with the σ-field of Borel sets and let P be the Lebesgue measure on [0,1]. Find E(X|Y) if: Homework Equations X(w)=5w^2 Y(w)= \left\{ \begin{array}{ll} 4 & \mbox{if $w \in [0,\frac{1}{4}]$} \\ 2 & \mbox{if $w \in (\frac{1}{4},1]$} \\ \end{array}...
33. ### Weird statement in my book about (measure theoretic) conditional expectation

My book tries to illustrate the conditional expectation for a random variable X(\omega) on a probability space (\Omega,\mathscr F,P) by asking me to consider the sigma-algebra \mathscr G = \{ \emptyset, \Omega \}, \mathscr G \subset \mathscr F. It then argues that E[X|\mathscr G] = E[X] (I'm...
34. ### Conditional Expectation of Sum

Hi everyone, I have a feeling the following property is true but I can't find it stated in any textbook/online reference. Maybe it's not true... Can someone verify/disprove this equation? E(A+B|C) = E(A|C) + E(B|C)
35. ### Conditional expectation of exponential random variable

For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value from searching in internet I found that E{X|X>a}=a+E{x} but I can not prove it Help please
36. ### Conditional expectation of exponential random variable

Homework Statement For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value Homework Equations The Attempt at a Solution
37. ### Conditional Expectation Question (Probability Theory)

Homework Statement (Question is #6 on p.171 in An Introduction to Probability and Statistics by Ruhatgi & Saleh) Let X have PMF Pλ{X=x} = λxe-λ/x!, x=0,1,2... and suppose that λ is a realization of a RV Λ with PDF f(λ)=e-λ, λ>0. Find E(e-Λ|X=1) The Attempt at a Solution The...
38. ### Tough Conditional Expectation Problem

Homework Statement Suppose that $Y$ is a random variable, $\mathcal{G}$ a $\sigma$-algebra, $E|Y| < \infty$. Show that $Y = E(Y|\mathcal{G})$ a.s. (a.s. = almost surely). Homework Equations We're given $Y$ integrable. The Attempt at a Solution It's recommended as a hint to prove...
39. ### Markov Chain Conditional Expectation

Hello, in relation to Markov chains, could you please clarify the following equations: In particular, could you please expand on why the first line is equal. Surely from , along with the first equation, this implies that: I just don't see why they are all equal. Please could you...
40. ### Stats - Conditional Expectation

Homework Statement [PLAIN]http://img222.imageshack.us/img222/2781/statsqk.jpg Homework Equations f_{X} (x) = \int^{\infty}_{-\infty} f_{X,Y} (x,y)\;dy f_{Y} (y) = \int^{\infty}_{-\infty} f_{X,Y} (x,y)\;dx f_{X|Y} (x|y) = \frac{f_{X,Y} (x,y)}{f_Y (y)} f_{Y|X} (y|x) =...
41. ### Fundamental question about conditional Expectation

Homework Statement I am familiar with the following kind of conditional expectation expression: \mathbb{E}[Y|X=x], where X and Y are random variables. I am wondering what the following conditional expectation stands for: \mathbb{E}[Y|X] How these two are related? How the second...
42. ### Conditional expectation of three exponential distributed r.v.

I've been struggling with this problem for more than 4 days now: Let A, B and C be exponential distributed random variables with parameters lambda_A, lambda_B and lambda_C, respectively. Calculate E [ B | A < B < C ] in terms of the lambda's. I always seem get an integral which is...
43. ### Conditional expectation on multiple variables

How to compute E[X|Y1,Y2]? Assume all random variables are discrete. I tried E[X|Y1,Y2] = \sum_x{x p(x|y1,y2) but I'm not sure how to compute p(x|y1,y2] = \frac{p(x \cap y1 \cap y2)}{p(y1 \cap y2)} If it is correct, how can I simplify the expression if Y1 and Y2 are iid?
44. ### Conditional expectation of exponential distribution.

I have been stuck at this calculation. There are two exponential distributions X and Y with mean 6 and 3 respectively. We need to find E[y-x|y>x] I keep getting it negative, which is clearly wrong. Anybody wants to try it?
45. ### Conditional expectation

Question 1) I have X and Y independent stoch. variables What is E[X^2 * Y | X] ? does it generally hold that if X and Y are independent, then every function of X (eg X^2) is independent of Y? Does E[X^2 * Y | X] then become E[X^2|X]*E[Y|X] = E[X^2|X]*E[Y] since X^2 is independent of...
46. ### Calculating Conditional Expectation for IID Normal Variables

If I have x1,x2 iid normal with N(0,1) and I want to find E(x1*x2 | x1 + x2 = x) Can I simply say: x1 = x - x2 and thus E(x1*x2 | x1 + x2 = x) = E[ (x - x2)*x2) = E[ (x * x2) - ((x2)^2) ] <=> x*E[x2] - E[x2^2] = 0 - 1 = -1?
47. ### Conditional expectation

Homework Statement An email is sent on the network in which the recipients (0,1,2,3,4,5} are in communication. 1 can send to 4 and 2 2 to 1,3,5 3 to 0,2,5 4 to 1, 5 5 to 0,2,4 0 to 3 and 5 If a message is sent to 2,3,4,5 it is forwarded randomly to a neighbour (even if this means a...
48. ### Conditional expectation and partitioning

Homework Statement I'm told that of n couples, each of whom have at least one child, with couples procreating independently and no limits on family size, births single and independent, and for the ith couple the probability of a boy is p_i and of a girl is q_i with p_i + q_i = 1. 1. Show...
49. ### Conditional expectation of Exp(theta)

Given X follows an exponential distribution \theta how could i show something like \operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta ? i have get the idea of using Memorylessness property here, but how can i combine the probabilty with the expectation? thanks. casper
50. ### Proving X=Y with Conditional Expectation

How can I do this? Let X,Y r.v., \mathbb{E}(X|Y)=Y and \mathbb{E}(Y|X)=X. Proove that X=Y a.s.