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
Given that $X$ is exponentially distributed continuous random variable $X\sim \exp(1)$ and $g(x)$ is as below. How can I find the Expectectaion of $g(x)$ for the condition that $x\geq Q$, i.e. $\mathbb{E}[g(x)\ | \ x\geq Q]$.
$$g(x) = \frac{A}{\exp(-bQ+c)}\Big(\frac{1 + \exp(-bQ+c)}{1 +...
Homework Statement
We have a coin with probability ##0\leqslant p \leqslant 1## of getting heads. We flip the coin until we get ##7## heads in a row. Let ##N_7## be the number of necessary flips to get the ##7## heads in a row.
What is the expected value ##E(N_7)##?
Homework Equations
The...
Homework Statement
Suppose that the number of eggs laid by a certain insect has a Poisson distribution with mean ##\lambda##. The probability that anyone egg hatches is ##p##. Assume that the eggs hatch independently of one another. Find the expected value of ##Y##, the total number of eggs...
Assume a Poisson process with rate ##\lambda##.
Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time...
Q The amount of time (in minutes) that an executive of a certain firm talks on the telephone is a random variable having the probability density:
$$f(x) = \begin{cases} \dfrac{x}{4}&\text{for $0 < x \le 2$}\\
\dfrac{4}{x^3}&\text{for $x > 2$}\\...
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...
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...
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...
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.
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...
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*}...
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...
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_{-∞}^{∞}...
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?
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...
(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...
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 +...
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...
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...
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...
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...
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)...
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
=...
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!
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...
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] =...
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...
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}...
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...
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)
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
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
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...
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...
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...
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...
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...
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?
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?
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...
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...
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...
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