All Questions
135
questions
4
votes
2
answers
128
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 ...
- 31
1
vote
1
answer
58
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 \...
- 31
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
42
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(...
- 127
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
51
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 $...
- 2,478
2
votes
1
answer
120
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 ...
- 5,420
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 ...
