Questions tagged [conditional-expectation]
A conditional expectation is the expectation of a random variable, given information on another variable or variables (mostly, by specifying their value).
6
votes
1answer
333 views
Calculating the expected value of truncated normal
Using the mills ratio result, let $X \sim N(\mu, \sigma^2)$, then
$E(X| X<\alpha) = \mu - \sigma\frac{\phi(\frac{a- \mu}{\sigma})}{\Phi(\frac{a-\mu}{\sigma})}$
However, when calculating it in R. ...
1
vote
0answers
12 views
Expected function evaluation of random variable w.r.t. different distribution
Suppose I have two continuous random variables on the same domain, $\xi \sim \mathbb{P}, \xi' \sim \mathbb{Q}, \in \Xi$ and joint probability $(\xi, \xi') \sim \Pi \in \Xi^2$ .
Now I would like to ...
3
votes
3answers
49 views
Iterated expectations and variances examples
Suppose we generate a random variable $X$ in the following way. First we flip a fair coin. If the coin is heads, take $X$ to have a $Unif(0,1)$ distribution. If the coin is tails, take $X$ to have a $...
0
votes
1answer
17 views
How does correlated relationship implies non zero conditional mean?
i.e. if $cov(X,Y)\neq0$, how do we derive $E(X|Y)\neq0$
it seems that it is quite obvious and usually be taken into granted,
but I just couldn't figure it out in a short time, can somebody give me a ...
0
votes
0answers
20 views
where can find good and in depth explanation for expectation, manipulation of expectation, sampling instead of expectation?
I look for a book or online source, for better understanding the expectation, expectation inside the expectation or sampling for calculation of expectation.
for example, in Richard S. Sutton's ...
0
votes
0answers
29 views
Conditional covariance of multivariate normal tail
Let $X\sim N(\mu,\Sigma)$, $t\in\mathbb{R}$, and $a$ be a non-zero vector of the same dimension as $X$. Define a random vector $Y=X\mathbb{1}(a^\top X\ge t)$, where $\mathbb{1}$ denotes the indicator ...
1
vote
1answer
47 views
Conditional Expectation of Log-Normal Distribution
I want to evaluate a conditional expectation of log-normal distribution. Let $y$ be a log-normal distributed random variable. So $\log(y)\sim N(\mu,\sigma^2)$. I want to calculate $E[y-1|y-1>0].$ ...
0
votes
0answers
19 views
combining variables from different tables
I'm having trouble combining variables from differents tables.
In this case, I have the exogenous variables $X$ (categorical) and the endogenous variable $Y$ (continuous). None of these variables are ...
2
votes
1answer
41 views
Conditional expectation; how to find E[xy] when E[x|y] is known?
In my studies for an exam which I have on Friday I have come across this assignment from last year in which the following question is asked:
"Let $E[x] = \mu$ and $var[x] = \sigma^2$. If $E[x \lvert ...
1
vote
1answer
37 views
What is the distribution of conditional expectation of a function f(X) of the random variable X? i.e. E(f(X)|X)
I have a continuous random variable X with a known PDF. I want to find the distribution of f(X) where ...
1
vote
1answer
52 views
Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$
I am unable to compute conditional probability(x|x>y) in the above question. Also, I am unable to determine the region of integration for calculation of the above expectation.
1
vote
3answers
48 views
Question Regarding Zero Conditional Mean
Hi I am a beginner to econometrics!
I have been dealing with bivariate regression. We use the formula $y = \beta_0 + \beta_1 x$.
I am told that if $E(u\mid x) \ne 0$ then the estimate of the slope ...
0
votes
0answers
15 views
Middle entries of a random vector - Conditional expectation and covariance matrix of normal distribution P(X2|X1, X3) [duplicate]
Let us consider the random vector $X=[X_1,X_2,X_3]$, which follows a multivariate normal distribution. That is, for each entry $X_i$: $X_i \sim N(\mu_i, \Sigma_i)$.
What I am trying to compute is the ...
1
vote
1answer
36 views
A very basic question on ENDOGENEITY
In the regression model
$Y$ = $\beta_0$ + $\beta_1X_1$ + $\beta_2X_2$ +.......+ $\beta_kX_k$ + $\epsilon$
where
$\epsilon$ = $\delta_0X_2$ + $\lambda$
Will this also be the case of endogeneity ...
0
votes
1answer
34 views
Conditional expectation setup
Suppose $\epsilon_0,\epsilon_1$ are iid random variables with density $f$ and cdf F and $c\in R$. Then why is:
$$E[\epsilon_1|\epsilon_1+c>\epsilon_0]=
\frac{\int_{-\infty}^{+\infty} x F(x+c) f(x) \...
0
votes
1answer
67 views
Law of Iterated Expectations Example
Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
8
votes
2answers
178 views
Conditional expectation of uniform random variable given order statistics
Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$.
How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
0
votes
1answer
46 views
Conditional expectation of two independent RV
The expectation of the product of two independent random variables $X$ and $Y$ is the product of the expectations:
\begin{align}
E(XY) = E(X)E(Y)
\end{align}
Let's add another random variable $Z$ in ...
1
vote
2answers
37 views
Is this formula for the Law of Iterated Expectations correct?
I saw two versions of the law of iterated expectations, this one:
\begin{align}
E(E(Y\vert X)) = E(Y)
\end{align}
and this one:
\begin{align}
E(E(Y\vert X_1, X_2)\vert X_1) = E(Y \vert X_1)
\end{align}...
4
votes
1answer
215 views
Can I use averages to improve the forecast of a multiple regression?
I have a cross-sectional multiple regression that I have estimated and now I would like to apply it to make a simple forecast of the dependent variable.
Take the data generating process
$$y_i =\...
0
votes
0answers
17 views
Conditional Expectation for Large Population
This was another question on my past exam and I'm curious to kmow how to solve it.
Let there be a population of 100,000 and the chance of someone being infected with a certain disease is 0.000001.
...
0
votes
0answers
30 views
What is the marginal variance of the mean of $y$ if I only know the variance of the conditional mean of $y$?
Suppose an estimate for the conditional mean of $y$ given $x$ is $\hat{E}(y|x)$. Suppose the variance (or the variance estimate) of $\hat{E}(y|x)$ is known to be $V(\hat{E}(y|x))$ for all $x$.
The ...
3
votes
1answer
87 views
Are the law of iterated expectation and the law of total expectations the same?
On the Wikipedia page of the Law of total expectations it is said that
The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, ...
0
votes
0answers
19 views
Question regarding conditional expectation [duplicate]
In Larry Wasseman's lecture notes(lecture 4, page 4) I found this statement
$\mathbb{E}[Y|X=x] = \sum_y y f_{Y|X}(y|x)$ or $=\int_y y f_{Y|X}(y|x)dy.$
An important point about the conditional ...
2
votes
2answers
52 views
Conditional expectation function
Consider the standard linear regression model given by
$Y = XB + \varepsilon$.
$E[Y\mid X] = XB$ if $E[\varepsilon \mid X] = 0$.
We say that the conditional expectation function is a random ...
4
votes
2answers
145 views
Different solution for a probability question
I got the following problem:
Find the probability that for two arbitrary numbers $x$ and $y$ with $x,y \in [0,1]$ they satisfy $x+y<1$ and $xy<\frac1{10}$.
In short words the sum of the two ...
0
votes
0answers
29 views
How to find conditional expectation$(X_1+X_2)^2$ given $X_1 = X_2$?
How do I show that $E[(X_1 + X_2)^2|X_1=X_2] = 2\sigma^2 + 4\mu^2$. When $X_1$ and $X_2$ follows $N(\mu,\sigma^2)$ independently.
As $X_1 = X_2$ is given, then I suppose I only need to find $E[4{X_1}^...
2
votes
1answer
35 views
Conditional-expectation operator inside of expectation operator
Let $b(\theta)$ be a parametric function, let $U$ be a sufficient statistic for $\theta$, let $T$ be an unbiased estimator for $b(\theta)$, and denote $g(U)$ as $g(u)=E[T|U=u]$. I am told that the ...
2
votes
2answers
97 views
Conditional Probability and Expectation for Poisson Process
To solve part (a) I have $P(X_2 = k\mid X_1 = 1)= \dfrac{P(X_2 = k \cap X_1 = 1)}{P(X_1 = 1)} = \dfrac{e^{-2}}{e^{-1}}=e^{-1}$. Then for part (b), for simplicity, I let $X_2=X$ and $X_1=Y$, then $$E(...
1
vote
1answer
52 views
Why is the conditional expectation the best predictor but only if we have the joint distribution?
If we want to predict one variable $Y$ based on another $X$, the best predictor is apparently $\mathbb{E}[Y \mid X = x]$. However, this apparently assumes two things:
The distribution is symmetric.
...
1
vote
0answers
92 views
Conditional probability of Negative Binomial R.V. given the SUM of its values
Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$.
How do you find the (expectation of) conditional probability ...
0
votes
1answer
22 views
Expectation conditional on self and others
I would simply like to know if:
$E[x_1|x_1,x_2]=E[x_1|x_2]$
or
$E[x_1|x_1,x_2]=E[x_1|x_1]=x_1$
or something completely different and why.
This is not homework. It came up because I'm trying to ...
0
votes
1answer
123 views
Expectation of standard exponential squared given sum of two standard exponentials
So I have been working on this question for a while and made some progress , but I run into a problem about the normalizing constant. The question is, for $X$ and $Y$ i.i.d. standard exponential, find ...
2
votes
1answer
76 views
Why does $\operatorname E(\varepsilon\mid x) = 0 \implies \operatorname{cov}(\varepsilon,x) = 0$?
I understand the intuition behind the question but I'm trying to prove it to myself with math.
0
votes
0answers
34 views
Expected value as an orthogonal projection
I'm reading a paper in which the expected value of a random variable, $\mathbb{E}[X]$, is characterized as an orthogonal projection. This is on page 10. I've seen the geometric interpretation of ...
1
vote
1answer
49 views
Conditional expectation of a vector
Suppose we have two random vectors $X=(X_1,X_2)^T$ and $Y=(Y_1,\dots,Y_n)^T$. I wish to find a simple definition or formula for
$$
E_{X|Y=y}[X]
$$
Intuitively, I think the following is correct:
$$
...
1
vote
1answer
37 views
Analytical solution to the covariance between a continuous and a categorical variable
Let $X$ be a continuous variable with mean $\mu$ and $Y$ be a categorical variable with event probability vector $\mathbf{p}$. I am trying to calculate $\operatorname{Cov}(X, Y)$.
I have the solution ...
3
votes
1answer
121 views
Multivariate Normal : expectation of X given Y is doubly-truncated
Let $(X, Y)$ be distributed as a multivariate normal with parameters
$$
\mu =
\begin{bmatrix}
\mu_X \\ \mu_Y
\end{bmatrix}
\qquad
\Sigma =
\begin{bmatrix}
\sigma_X^2 & \sigma_{XY} \\ \sigma_{XY} &...
1
vote
1answer
50 views
Independence and conditional distribution
In a problem that I'm solving I find that:
"Let data (yi,xi) be sampled randomly from a two-dimensional distribution such that y|x is N(ɑ,x^2σ^2)".
Are y and x i.i.d? maybe just identically ...
1
vote
1answer
26 views
Difference between averaging and ignoring the partial dependencies?
This question sparks from model interpretation/visualization. To graph the dependency of a function with >2 arguments, one often needs to ignore or average out some arguments.
Problem set
Hastie, ...
1
vote
1answer
80 views
Proof of contemporaneous exogeneity, and its implications for an AR(1) model
It can be shown by contradiction that exogeneity fails to hold for an AR(1) model.
Is there any proof that contemporaneous exogeneity does not fail to hold?
All I've come across is assuming it does ...
4
votes
1answer
53 views
Conditional mean and co-variance in $VAR(p)$ conditional on one lag only
Suppose I have a $p$'th order vector auto regression
$$\vec Z_t = F_1\vec Z_{t-1}+F_2\vec Z_{t-2} + \cdots +F_p \vec Z_{t - p}
+ \vec \epsilon_t,\qquad \vec\epsilon_t\sim N_q(\vec0,Q)$$
where $...
1
vote
0answers
56 views
Understand conditional expectation w.r.t. sigma -algebra [duplicate]
When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is
$$
E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
1
vote
1answer
54 views
How to deal with expected value in the context of time series?
For example, in this MA(2) model,
$y_t = u_t + \phi u_{t-2}$
$u_t$ is identically, independently, normally distributed with a mean of 0 and a variance of $\sigma^2$. (Does variance matter here?)
I ...
0
votes
0answers
38 views
Questions relating the definition of conditional expectation $E[g(X)|M]$ where is $X$ is random variable and M is an event
I saw the following definition of conditional expectation from a book:
if M is event and X is continuous random variable then we define:
$$E[X|M]=\int_{-\infty}^\infty xf(x|M)dx$$
Which is the ...
6
votes
1answer
121 views
Estimate E(Y|X,Z) from E(Y|X) and E[Y|Z]
Can I estimate $E[Y|X, Z]$ if I know $E[Y|X]$ and $E[Y|Z]$?
As an example, let's say I have a model where
$X\sim N(0, 1)$,
$Y = aX+\epsilon_Y, \epsilon_Y \sim N(0, 1)$,
$Z = bY+\epsilon_z, \...
0
votes
0answers
22 views
Calculate target variance contionaly to continuous variable
I'm trying to get the conditional variance of my target Y conditional to my inputs X. To do this I first discretized my inputs variable into 1 qualitativ variable and then I computed the variance of Y ...
2
votes
2answers
45 views
Generate Data where outcome is conditional on independent variables [closed]
I want to generate a synthetic dataset {Y, X1, X2}. Independent random variables X1 and X2 follow bernouli distribution where probabilities for X1 and X2 are known.
Whereas, outcome variable Y needs ...
-1
votes
1answer
37 views
Prove convergence in distribution, probability, or quadratic mean for a sequence of binary variables that depend on another binary variable
Suppose that $X$ has the support set $\{1, -1\}$, and $P(X = 1) = P(X = -1) = 0.5$.
Suppose that $X_n$ has the support set $\{X, e^n\}$, and
$P(X_n = X) = 1 - \frac{1}{n}$
$P(X_n = e^n) = \frac{1}...
2
votes
0answers
99 views
Proving covariance equals zero given a specific conditional expectation
I'm trying to prove the following:
Given $𝐸[𝑋|𝑌 = 𝛽] = 𝐸[𝑋]$ for any value of $\beta$, prove that $\operatorname{Cov}(𝑋,𝑌) = 0$;
So I was thinking to start with the definition of $\...
