All Questions
135
questions
4
votes
2
answers
130
views
Calculating a Conditional expectation
My question is the following. Given that we have $n$ i.i.d. random variables $X_1,...,X_n$ with distribution $f(x)=\frac{2}{\lambda^2}x\mathbf{1}_{[0,\lambda]}(x)$, where $\lambda> 0$ is some ...
- 53
0
votes
2
answers
102
views
Show $E[Y | E[Y | X]] = E[Y | X]$. [closed]
Given Let $X$, $Y$ be random variables on a common probability space $(\Omega, \mathcal{A}, P)$.
To show: $E[Y | E[Y | X]] = E[Y | X]$
For this problem, I'm unsure how to rewrite the left-hand side of ...
- 762
2
votes
1
answer
51
views
Show: $\text{Var}(Y) = \text{Var}(Y - E[Y | X]) + \text{Var}(E[Y | X])$.
Given: Let $X$, $Y$ be random variables on a common probability space $(\Omega, \mathcal{A}, P)$.
To prove: $\text{Var}(Y) = \text{Var}(Y - E[Y | X]) + \text{Var}(E[Y | X])$.
Attempt:
$\text{Var}(Y) =...
- 762
0
votes
0
answers
43
views
conditional expectation of non-negative variable
In Angrist's book Mostly Harmless Econometrics, section 3.4.2, it says
$$ E[Y_i|D_i]=E[Y_i|Y_i>0,D_i]P[Y_i>0|D_i] $$, where $Y_i$ is a non-negative variable ($Y_i$ can be $0$) and $D_i$ is a ...
- 53
0
votes
1
answer
36
views
Conditional probability involving two random times, where only the distribution of one of them is used.
Consider two random times $\tau_{1}$ and $\tau_{2}$ defined on a common probability space $\left(\Omega, \mathcal{G}, \mathbb{Q}\right)$ with $\mathbb{Q}(\tau_{k}=0)=0$ and $\mathbb{Q}(\tau_{k}>t)&...
- 91
0
votes
1
answer
26
views
A question on conditional expectation of a random variable
Consider the joint probability density function:
$$f(x_1,x_2)=
\begin{cases}
2e^{-2x_1}, & \text{ for } 0 \le x_2 \le x_1 < \infty \\
0, & \text{ elsewhere} \\
\end{cases}
$$
Find the ...
- 779
0
votes
1
answer
39
views
Calculate the Variance of $\min(N_k,p)$
I am trying to compute the variance of a random variable $\min(N_k,p)$, where $N_k$ is a random variable and $p$ is a fixed number. I have computed the expectation ...
- 33
1
vote
1
answer
59
views
Find the conditional expectation $E[X \mid X \leq p]$
Let $X$ be a discrete random variable that can take values from 1 to $n$, where $n$ is a large fixed number and also let $p\ll n$ be a fixed number. I am trying to find the expectation $$E[X \mid X \...
- 33
0
votes
0
answers
60
views
Conditional expectation on continuos random variable with zero density obtained from non-zero density variables.
Let $Y$, $X_1$ and $X_2$ be three continuous real random variable with $f(x_1, x_2) >0$ everywhere on $R^2$ and denote by $g(x_1, x_2) = E[Y|X_1 = x_1, X_2 = x_2]$. Then $g(0,0) = E[Y|X_1 = 0, X_2 =...
- 45
0
votes
0
answers
43
views
Changing variables inside conditional expectation
Say we have a random variable $X$ on a probability space, taking values in the natural numbers. We want to compute $E(X)$. Letting $n\in \mathbb{N}$ and using the law of total expectation, $E(X) = E(E(...
- 135
1
vote
1
answer
36
views
Finding unconditional expectation using iterated expectation [closed]
Discrete random variable $\Theta$ is uniformly distributed between 1 and 100. Given $\Theta$ discrete random variable $X$ is uniformly distributed between 1 and $\Theta$. Show that
$E[X^2] = \frac{1}{...
0
votes
0
answers
52
views
Computing $\mathbb{E}[X \mid X \land t]$ for exponential$(1)$ random variable $X$ [duplicate]
Let $X$ be an exponential$(1)$ random variable defined on a probability space $(\Omega,F,P)$. That is, for any $a \geqslant 0$:
\begin{equation*}
P \{ X \leqslant a \} = 1-e^{-a}
\end{equation*}
Fix $...
- 2,497
2
votes
1
answer
122
views
Existence proof of conditional expectation
I am self-learning introductory stochastic calculus from the text A first course in Stochastic Calculus, by Louis Pierre Arguin. I'm struggling to understand a particular step in the proof, and I ...
- 5,450
2
votes
1
answer
86
views
If $E|Y|\lt \infty$, $E[Y|X] = m(X)$ and $(X_1,Y_1)$ come from same distribution as $(X,Y)$. Is it true that $E[Y_1|X_1] = m(X_1)$?
If $X,Y$ are random variables, such that $\mathbb E|Y|\lt \infty$, $\mathbb E[Y|X] = m(X)$ where $m$ is some Borel measurable function and $(X_1,Y_1)$ come from same distribution as $(X,Y)$. Is it ...
- 798
0
votes
1
answer
40
views
Expectation Conditioned on $\sigma$-subalgebra
In preparing for my upcoming qualifying exam, I have encountered the following problem:
Consider the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ where $\Omega = [0,1]$, $\mathcal{F}$ is the ...
0
votes
1
answer
61
views
A question about the geometric interpretation of the conditional expectation
Let $X$ and $Y$ be two random variable. Suppose $E[X|Y]=0$, we have
$$E[XY]= E[\, E[ XY |Y ]\, ] = E[\, Y E[ X |Y ]\, ]= E[0] = 0$$
A geometric intuition also corroborates this fact if we think $E[X|...
- 351
5
votes
1
answer
132
views
Sigma-algebra generated by conditional expectation
I am dealing with the following question: given two dependent random variables $X_1,X_2$, I am wondering whether the following equivalence for the generated sigma-algebras holds:
$$\sigma(X_1)=\sigma(...
- 95
1
vote
0
answers
97
views
Find the conditional moments of a random variable
Suppose that we have three different variables $x$, $y$ and $z$, where $x$ stands for the state of the world and $\mathbb{X}$ is the state space, such that $x\in\mathbb{X}$. The following information ...
- 257
1
vote
1
answer
117
views
Elementary explanation of getting two consecutive $6$'s in a die roll experiment
I know that there are already numerous questions that adress this problem. However, I am not interested in a soltuion at all but in an explanation of a particular solution (see https://math....
- 4,564
2
votes
0
answers
112
views
Conditional expectation of typos
I'm wondering how one could solve this problem:
A text consists of $n$ characters, each of which is a typo with probability $p$ (independently).
A proof reader then reads through the text and ...
2
votes
0
answers
420
views
If $X$ is independent of $\mathcal{G}$ and $Y$ is $\mathcal{G}$-measurable, then $\mathbb{E}(\varphi(X, Y) | \mathcal{G}) = \mathbb{E}(\varphi(X, Y))$
I'm reading a proposition given without proof in this note.
Proposition 12.4. Let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}, X, Y$ be two random variables such that $X$ is independent of $...
- 5,817
7
votes
4
answers
905
views
Why is $E[X|X+Y] = E[Y |X+Y]$ if X,Y are i.i.d random variables
In proof of the fact that $E[X|X+Y] = \frac{X+Y}{2}$ when $X,Y$ are independent, identically distributed random variables, one uses the observation that $E[X|X+Y] = E[Y|X+Y]$ but I don't see why this ...
- 117
4
votes
1
answer
320
views
Doob's Optional Stopping Theorem to find probabilities of stopping times
Suppose we have a simple random walk starting from $S_0=0$, and $S_n=X_1+\dots+X_n$ such that $$\mathbb{P}(X_i=1)=p \hspace{1em}\mathbb{P}(X_i=0)=r \hspace{1em} \mathbb{P}(X_i=-1)=q$$ for positive $p,...
- 115
0
votes
1
answer
157
views
Substitute in what is known in conditional expectation
Motivation:
It might appear intuitive that $E(f(X,Y)|Y=y)=E(f(X,y)|Y=y)$, i.e. we just substitute in what is known in the conditional expectation. However, I want to prove this rigorously using the ...
- 167
0
votes
0
answers
279
views
Definition of conditional expectation given an event
The conditional expectation of a random variable $X$ given a nonnegligible event $A$ is usually defined as $\mathbb{E}(X\mathbb{1}_A)/\mathbb{P}(A)$. How does one derive the fact that $\mathbb{E}(X|A)=...
- 1,022
0
votes
1
answer
210
views
Property of conditional expectation $E(X | \mathcal{V})$ where $\mathcal{V}$ is $\sigma$-algebra.
I'm self-studying the probability theory, and I got stuck on the understanding of the definition given below and some consequences that follow from that definition.
Let $(\Omega, \mathcal{U}, P)$ be a ...
- 337
2
votes
1
answer
52
views
Showing expectation of a finite sum of a sequence of random variables, squared
I am working with Loeve's "On Almost Sure Convergence", specifically on the extension of Kolmogorov's inequality in Lemma 5.1.
As part of the proof, with the assumption $E(X_n|X_{n-1},...,...
- 35
3
votes
1
answer
62
views
Equivalence of conditional expectations wrt two sigma-algebras
Let $X$ be a random variable and let $\mathcal{G}$, $\mathcal{H}$ be two sub-$\sigma$-algebras. Consider the equation $$\mathbb{E}(\mathbb{E}(X|\mathcal{G})|\mathcal{H}) = \mathbb{E}(X|\mathcal{G}\cap\...
- 1,023
1
vote
0
answers
41
views
Independence of random variables meas. wrt independent sigma algebras.
Suppose we have two independent $\sigma$-algebras, $\mathcal{G}$ and $\mathcal{H}$. Let $X$ and $Y$ be two $(\mathcal{G}\cap\mathcal{H})$-measurable random variables. Then $\sigma(X)\subseteq\mathcal{...
- 1,023
3
votes
1
answer
41
views
Find Conditional expectation of uniform variables ...
Let $\xi,\eta$ be independent random variables, both with uniform distribution on $[0,2]$. Find $E[\eta^2|\xi/\eta]$.
My attempt to solve the problem is in the attached file.
I believe I solved it, ...
1
vote
0
answers
39
views
How does conditional expectation tell the average of r.v. X on the union of some basic events?
Let $X\in L^1$ be a $\mathcal F$-measurable random variable and $\mathcal G$ be a sub σ-algebra of $\mathcal F$. We say a $\mathcal G$-measurable random variable $\mathbb E[X|\mathcal G]\in L^1$ is ...
- 57
1
vote
1
answer
136
views
How does one compute conditional expectation with respect to a continuous random variable?
I can't wrap my head around the way to compute conditional expectation with respect to a continuous random variable. For instance, consider a probability space $(\Omega, A, P)$, where $\Omega = [0,1]$ ...
- 263
1
vote
1
answer
343
views
Proving conditional expectation is bounded if expectation is bounded
Suppose we have a random variable $X$ bounded in expectation, i.e., $E[X] < c$.
If $X \geq 0$, then is it correct to infer that the conditional expectation of $X$ given another r.v. $Y=y$ is ...
- 219
0
votes
1
answer
128
views
Conditional Expectation of Function of I.I.D Random Variables [duplicate]
Let X,Y,Z $\stackrel{i.i.d}{\sim}$ N($0$,$1$). I am supposed to find the E($2$X+$3$Y | X+$3$Y-Z =$4$)
I Tried to solve the problem by considering A= $2X+3Y$ ~ $N(5,5)$ and B=X+$3$Y-Z ~ $N(3,5)$ but ...
- 119
2
votes
3
answers
263
views
What is the distribution of $E[X\mid Y]$?
Let $(X, Y)$ be two r.v. with joint p.m.f. described by the following table
What is the marginal distribution of $X$?
Are $X$ and $Y$ independent?
What is the conditional p.m.f. of $X$ given $Y=0$?
...
- 1,583
10
votes
1
answer
392
views
What am I writing when I write $\mathbf X \mid \mathbf Y$?
Suppose $\mathbf X$ is a random variable and $A$ is an event in the same probability space $(\Omega, \mathcal F, \Pr)$. (Formally, $\mathbf X$ is a function on $\Omega$, say $\Omega \to \mathbb R$; $A$...
- 145k
0
votes
0
answers
145
views
Range of conditional expectation
Let $X$ be a random variable with values in $A$.
Let $Y$ be random variable with values in $B$.
Let $g: A\times\mathbb R \to C$ a differentiable function.
Define $h(z)= \mathbb E(g(X,z)\mid Y=y)$.
...
- 73
1
vote
2
answers
74
views
What is the expectation of $(\langle x, w \rangle - \langle y, w \rangle)^2$, where $x,y,w$ are independent Bernoulli random vectors?
I'm stuck on the following problem.
Consider three independent Bernoulli random vectors $x,y,w$ of length $n$, where each entry follows the Bernoulli distribution $B$ with $P(B=0)=P(B=1)=\frac{1}{2}$.
...
- 37
1
vote
1
answer
113
views
$X=\mathbb{E}[X\mid\mathcal{F}] $ implying $\mathcal{F}$-measurability of $X$
For a random variable $X$ we know, that if $X$ is measurable w.r.t some filtration $\mathcal{F}$ it satisfies $X=\mathbb{E}[X\mid\mathcal{F}]$.
I wonder whether one can reverse this property and state,...
- 543
3
votes
1
answer
159
views
Effect on expected value of conditioning on inequality between random variables (do we have E[X | X>S] ≥ E[X | X>S, Y>S]?)
I've been trying to prove the following inequality:
$$
\mathbb{E}[X \mid X > S] \geq \mathbb{E}[X \mid X>S, Y>S]
$$
where $X$, $Y$, $S$ are mutually independent real-valued random variables ...
- 31
1
vote
1
answer
957
views
Proof Linearity of Conditional expectation
How we can proof that:
$E[X - Y|W] = E[X|W]-E[Y|W]$
I try to use the definition of Conditional Expectation:
$E[X|Y=y]= \sum_x \cdot p(x|y)$ and then to substitute $X=A-B$
Is it the right way? And ...
user840425
11
votes
1
answer
1k
views
Does almost sure convergence and $L^1$-convergence imply almost sure convergence of the conditional expectation?
Question. Let $ X_{n}, X $ be random variables on some probability space $ ( \Omega, \mathcal{F},\mathbb{P} ) $ and let $ \mathcal{G} \subset \mathcal{F} $ be a sub-$\sigma$-algebra. Moreover ...
- 115
4
votes
1
answer
320
views
Approximating $E[g(Y)]$ with a Taylor series is easy, but $E[g(YX)]$ seems much tougher (potentially impossible)?
Let $X$ and $Y$ be continuous random variables with $Y \sim \mathcal{N}(\mu_Y,\sigma_Y^2)$.
I want to approximate $E[g(YX)]$ where $g(x) = e^{-x^2}$.
Specifically, my plan is to use a Taylor expansion ...
- 10k
2
votes
3
answers
256
views
How to show $E[X\mid X = x] =x$?
I read that $E[X\mid X = x] =x$ but I don't get that when I try to prove it:
\begin{align}
E[X\mid X = x] &= \sum x P(X=x|X=x) \\
&= \sum x \frac{P(X=x,X=x)}{P(X=x)} \\
&= \sum x \frac{P(X=...
- 165
0
votes
1
answer
27
views
Question about conditioning on sequences of random variables, where the sequence is related by a function $f$
Let $A_k, B_k$ be two sequences of random variables, where $A_k = f(B_k)$
Then is it true that
$$\mathbb{E}[A_{k+1}|A_0, \ldots, A_k] = \mathbb{E}[f(B_{k+1})|B_0, \ldots, B_k]$$
If so, what is the ...
- 557
3
votes
0
answers
77
views
Correlation of conditional expectation of uncorrelated random variables
Let $X,Y\in\mathcal{L}_{2}\left(\Omega,\mathcal{F},\mathbb{P}\right)$ satisfy $\mathbb{E}\left[X\right]=\mathbb{E}\left[Y\right]=\mathbb{E}\left[XY\right]=0$, and $\mathbb{E}\left[X^2\right]=\mathbb{E}...
- 191
0
votes
2
answers
54
views
Does expectation inequality imply conditional expectation inequality?
Given a probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$ and two random variables defined on it, does it hold true that
$$
\mathbb{E}\left(X\right)<\mathbb{E}\left(Y\...
0
votes
1
answer
41
views
Is it true that $\mathbb{E}\{X\}=\mathbb{E}\{Y\}$ $\Rightarrow$ $\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\mathcal{F}\}$? [closed]
Given $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and two r.v.'s $X$ and $Y$ defined on it, does it hold true that:
$$\mathbb{E}\{X\}=\mathbb{E}\{Y\}\Rightarrow\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\...
- 1
1
vote
1
answer
277
views
Conditional expectation for independent random variables
I have a question about the conditional expectation with some independence conditions for random variables and $\sigma$-fields.
For a random variable $X$ with $E|X| < \infty $, if $Y_1$ and $ ...
- 27
4
votes
1
answer
224
views
Law of large numbers holding uniformly with respect to a distribution
Let $X$ and $\varepsilon$ be independent random vectors, $\mathcal{X} = \text{supp}(X)$, and $Y = f(X) + \varepsilon$ for some function $f$.
For any $x \in \mathcal{X}$, let $y^i = y^i(\omega)$, $i \...
- 1,788