A conditional expectation is the expectation of a random variable, given information on another variable or variables (mostly, by specifying their value).

learn more… | top users | synonyms

1
vote
0answers
8 views

Is the maximum entropy distribution the same for conditional and unconditional moments?

Suppose I have a set of observations drawn from some finite interval from a distribution that has a range that includes that interval but extends beyond it. This could be either because of the values ...
0
votes
1answer
15 views

Conditional expectation notation

I am learning about Heckman selection model and confused with conditional expectation notation. If the equation that determines sample selection is: The outcome equation is: If error terms have ...
4
votes
0answers
19 views

Get covariance from conditional covariance for lognormal (and other) observations?

Consider lognormal random variables $X_1$ and $X_2$ with correlation coefficient $ρ$ and a partial observation sample of them of length N, the sample being partial because it only contains occurrences ...
1
vote
1answer
44 views

Can we calculate $E[Y|Z]$ if we know $f(Y|X)$ and $f(X|Z)$?

While reading Jennifer Hill (2011), p. 220, in the context of conditional average treatment effect, the author is able to calculate $E[Y|Z]$ from $f(Y|X)$ and $f(X|Z)$. My attempt to replicate the ...
3
votes
1answer
42 views

Conditional expectation of a univariate Gaussian

Suppose I have a univariate Gaussian distribution with mean $\mu_X$ and standard deviation $\sigma_X$, and I know the random variable $X$ is least some positive value $y$: $X \geq y$. What is the ...
0
votes
0answers
25 views

When do we have that $E[X_{T+1} | X_T] = X_T$?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$: Let $X = \{X_n\}_{n \in \mathbb N}$ be a $(\{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{...
1
vote
1answer
18 views

Minimizing MMSE over positive random variables

Let X be a random variable with a finite second moment. We know that: Argmin E(X-Y)^2 = E(X|g), Where the minimum is taken over all g-measurable random variables Y. How can I find argmin E(X-Y)^2 ...
3
votes
1answer
53 views

Expectation of two identical lognormal distributions

I would like to compute the conditional expectation (on an interval from $c$ to $\infty$) of the minimum of two log normal distributions. Denote $X_1$, $X_2 \sim LN(0, \sigma)$, the associated ...
0
votes
1answer
24 views

expectation of conditional expectation

Given $(X,Y)$, 2-dimensional probability vector, and let $g: R^2 \rightarrow R, E[g(X,Y)^2 ] < \infty$ and $h:R \rightarrow R, E[h(X)^2] < \infty $, prove the following: $$E[h(X)\{g(X,Y)-E[g(X,...
0
votes
1answer
40 views

Example of Conditional Expectation in Markov Chain

My question is from the book Introduction to Probability Models, 10th edition, by Sheldon Ross. Here is a example in the book. Consider a Markov chain with states $0, 1,\cdots , n$ having $P_{0,1} = ...
0
votes
0answers
6 views

question about the conditional error term in linear regression

Suppose we are given $n$ i.i.d. random vectors $\{ y_i,X_i \}$ where $y_i$ is a random scalar and $X_i$ is a random vector. Further suppose that $\epsilon_i$ is a linear function of $\{ y_i,X_i \}$. ...
4
votes
0answers
31 views

Asymptotic conditional expectation

Problem Setup Let $\{X^d_1, X^d_2, \cdots, X^d_n\}$ be a $d-$dimensional zero-mean, i.i.d. random variables. Let $S_n^d$ be $$ S^d_n = \frac{\sum_{i=1}^n X_i^d}{\sqrt{n}} $$ Let $Y^d$ be a zero-...
2
votes
2answers
56 views

Conditional Expectation of a function and another Conditional Expectation

Suppose $\{X_1,X_2,Z\}$ is a vector of 3 real valued continuous random variables with compact support, $f_1(X_1,X_2)$, $f_2(X_1,X_2)$, and $g(X_1,X_2)$ are measurable functions with at least 2 ...
0
votes
0answers
8 views

Coordinate Ascent for Variational Inference: Deriving Updates

I am working with the following model and am attempting to derivate coordinate ascent updates using mean field variational inference: Sample $p_X \sim Beta(\alpha_1, \alpha_2)$ Sample $p_Y \sim Beta(...
4
votes
1answer
39 views

Conditional expectation for non-gaussian variables

Let $A$, $B$ be two zero-mean random variables. Let the variance be $\sigma^2_A$, $\sigma^2_B$ and let the correlation be $\sigma_{AB}$. Consider the following expression :- $$ \mathbb{E}\big[A|B=b\...
2
votes
1answer
16 views

Deviation due to conditioning

Let $A$ and $B$ be random variables. Can we upper-bound the following expression? $$ \mathbb{E}\Big[\Big(\mathbb{E}[A|B] - \mathbb{E}[A]\Big)^2\Big] $$ The above looks classical research. However, I ...
0
votes
2answers
57 views

Zero conditional mean assumption (how can in not hold?)

Zero conditional mean of the error term is one of the key conditions for the regression coefficients to be unbiased. My question is: how can this assumption at all be violated if errors are equal to ...
3
votes
1answer
70 views

Estimating a function $f$ of a random vector $\mathbf{x}$ by a subset of the coordinates of $\mathbf{x}$ after a rotation of the input space

Suppose I have $$h=f(\mathbf{x})$$ with $f$ a deterministic function and $\mathbf{x}=(x_1,\ldots,x_n)$ a random vector of known distribution. I'm not using the capital letter notation for random ...
0
votes
0answers
21 views

Variance of the difference of two random variables compared to the difference of conditional expectation

Fix a probability space $(\Omega,\mathscr{F},\mathbb{P})$. Let $Y$ be a square integrable random variable $\mathbb{E}Y^2 < \infty$ and let $\mathscr{G}$ be a sub-$\sigma$-algebra of $\mathscr{F}$. ...
1
vote
1answer
50 views

Existence of the conditional tail mean

Does the existence of the first moment of a generalized Pareto distribution with support $[0,\infty)$ imply the existence (finiteness) of the conditional tail mean -- i.e. what in risk management is ...
1
vote
0answers
26 views

Finding conditional expected value

Given that X and Y are two independent exponentially distributed random variables with parameters a and b respectively. let Z = max(X,Y) find E[X|Z] attempt: I found that: P(Z=X) = b/(a+b) and P(...
0
votes
0answers
7 views

Adding dependent random normal distributions and conditional expectations

My problem is as follows: Let $X, B_1, B_2, B_3$ be independent normal random variables with $\mu = 0$, $\sigma = 1$. Let: $Y_1 = X + B_1$ $Y_2 = 2X + B_2$ $Z = X + B_3$ Then, I had to find $Z' = ...
0
votes
1answer
33 views

Confused about “iterated expectations” step in a derivation

Let $X$ and $Y$ be jointly normal random variables with mean zero, variances $\sigma_x^2$ and $\sigma_y^2$, correlation $\rho$. Then, given a constant $\delta$: $\mathrm{E}\left[x\ |\ y>\delta\...
0
votes
0answers
25 views

Approximating the conditional expectation in simulations

I am simulating stock returns, which are governed by the following equations $r_t = \mu + \delta r_{t-1} + \varepsilon_t$ $\sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma^2_{t-1}$ $\...
1
vote
0answers
32 views

Expectation from conditional expectations

Consider a real-valued random variable $X$ and a dichotomus random variable $Y\sim Be(p)$, all defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Could you help me to show ...
0
votes
0answers
23 views

Expectation of the distant future values conditional on the current information today

I am trying to understand a structural econometric model, in particular the one presented in "Stock J. H. and Wise,D. A., 1990. Pension, the option value of work, and retirement, Econometrica, 58 (5), ...
1
vote
1answer
44 views

I know $Y|Z \sim\mathbb{N}(Z,\sigma^2)$, but what is $Z|Y$?

Say I know the conditional distribution: $Y|Z \sim\mathbb{N}(Z,\sigma^2)$ Now, what if I reversed this though and wanted to find the conditional distribution $Z|Y$? From intuition I would expect ...
1
vote
0answers
32 views

Average conditional variance

Related to Explanatory power of variable. Given a data for 1d variable $Y$ and multidimensional variable $X$, what is the best way to compute average conditional variance of $Y$ given $X$: $$ \Bbb V(...
3
votes
0answers
44 views

Expectation of a conditional density

I'm trying to figure out why the following equation holds: $$f_{Y}(y) = E(f_{Y|X}(y|X))$$ I have sort of "worked out" the RHS to be: \begin{align} f_{Y}(y) &= E(f_{Y|X}(y|X)) \\[5pt] &...
3
votes
1answer
88 views

Conditional Mean in Linear Regression

I have a question about linear regression in general. Suppose we have the following Data Generating Process:$$y_{i}=x_{i}\beta+\epsilon_{i}$$ Now, the thing is that from my understanding, each ...
0
votes
0answers
15 views

Deducing the regression function using the Squared Error Loss Function [duplicate]

I am reading Elements of Statistical Learning, and came across a deduction which I cannot understand. In the second chapter, the author defines the squared error loss and deduces the conditional ...
1
vote
2answers
80 views

Can I apply statistics to catch rare diseases or make decisions about fate?

Suppose I am 50 years old and a study found that in my city 10 people (CI=4 people) die at 50. Now it's July and I have a very malignant disseminated cancer and this year 14 people died at age 50 in ...
3
votes
1answer
50 views

Conditional Expectation of sum of uniform random variables?

Let $X,Y$ be independent uniform random variables on interval $[0,1]$. Can someone show me how to find the expectation of $X$ conditioned on $X+Y \ge (\text{say}) 1.3$? $$E[X | (X+Y) \ge 1.3]$$ ...
4
votes
1answer
53 views

X is stochastically increasing in Y $\implies$ $E\left[Y| X\right]$ increasing in X

I have two random variables, $X$ and $Y$. I know: $\text{pr}\left(X \le u | Y\right)$ is a decreasing function of $Y$ for all $u$. Does this imply that: $\mathbb{E}\left[Y | X\right]$ is increasing ...
0
votes
0answers
30 views

With given numbers $a_1, a_2,a_3,…,a_N$, let $W=\Sigma_{i \in s_n}a_i$. Calculate the mean and variance of $W$

From the set $R=\{1,2,3,...,N \}$, a set $s_n$ of $n$ numbers are chosen without replacement, $0<n<N$. With given numbers $a_1, a_2,a_3,...,a_N$, let $W=\Sigma_{i \in s_n}a_i$. Calculate the ...
1
vote
1answer
54 views

Consistent estimator of the expectation of a conditional probability

I'm stuck in a problem where I have distribution distribution $P(\boldsymbol{x})$, from which I know how to sample from (i.i.d.) and two functions of the random variable $\boldsymbol{x}$: $E(\...
0
votes
0answers
36 views

Does the LHS of $E[X_n | \mathscr F_{n-1}]$ make sense even if $X_n$ is not integrable or adapted?

Let $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ be a filtered probability space. Then $X_n$ is a $(\{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)-$martingale if: $X_n$'s ...
2
votes
2answers
163 views

Expectation and Conditional Independence

This question is an aside from another question here on CV. We know that the expectation of the product of two independent random variables is the product of expectations, i.e., $$\mathbb{E}[XY]=\...
0
votes
0answers
17 views

Estimating the conditional expectation

Consider the discrete random variables $G,W,N$ all defined in the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, respectively with support $\mathcal{G}, \mathcal{W}, \mathcal{N}$. Consider ...
0
votes
0answers
74 views

Decomposition property of conditional expectation function

Let $Y$ be the outcome variable and $X$ and $U$ be covariates. The well known decomposition property of CEF is given by: \begin{align*} Y=E(Y|X,U)+e \end{align*} where \begin{align*} E(e|X,U)=0 \end{...
2
votes
2answers
100 views

Prove $E[|Z|] < \infty \to E[|Z_n|] < \infty$

Let $Z$ be an integrable random variable on filtered probability space $(\Omega , \mathscr F, (\mathscr {F_n})_{\{n \in \mathbb{N}\}}, \mathbb P)$ Define $Z_{n} := E[Z|\mathscr {F_n}]$. Show that $...
0
votes
0answers
32 views

Conditional covariance of AR(1)

I am trying to derive the conditional covariance of an AR(1) process, $\text{Cov}(y_t,y_{t+h})$. I have been trying to solve it taking in account the law of iterated expectations (LIE). However, I ...
1
vote
1answer
60 views

Expected Value in Poisson Point Process with Prior Knowledge

I have a setup with a homogeneous Poisson Point Process (PPP) of intensity $\lambda$ in $W \subseteq \mathbb{R}^d$ and a set $A \subseteq W$. I'm looking for the expected value of points in set $A$, ...
0
votes
1answer
32 views

Identifying Assumption D-I-D Conditional Expectation of Errors

I am reading this paper about Difference in Difference and I am confused by the statement regarding the conditional expectation of the error. I dont think I ever quite grasped what this value ($E[\...
0
votes
0answers
60 views

Poisson Processes: Conditional Expectation

For a Poisson process of an event with some rate r what is the probability to have an occurrence of this event in the next time interval given that n occurrences of this event happened in the k ...
2
votes
0answers
77 views

Conditional Variance Representation - Correct Form [closed]

Which of the below two representations of conditional variance is correct? Here, $X$ and $Y$ are random variables. ONE $$\text{Var}\left\{ Y|X\right\} =E\left[\left\{ Y|X\right\} ^{2}\right]-\left[E\...
1
vote
1answer
43 views

Mean conditional on filtration

I'm quite new to time series analysis. I'm reading a text book, where the conditional expected value of a process $(X_t)$ is written as $E[X_t|\mathcal{F}_{t-1}]$, where $\mathcal{F}_{t-1}$ is called ...
3
votes
2answers
168 views

simple conditional expectation question

I came across the following in an article that I was reading. I cannot prove to myself that it is true or not? Any help would be greatly appreciated. Is is true that $E[x|y]=\rho y$, $E[x|z]=0$ ...
1
vote
0answers
66 views

Derivation of conditional expectation and variance of the AR(1) process

I have a question regarding the AR(1) process. I want to derive the conditional expectation $E(X_{(t)}| X_{(0)})$ and the variance $\operatorname{Var}(X_{(t)}|X_{(0)})$ of the AR(1) process: $$X(t)=aX(...
0
votes
0answers
20 views

Does this type of diagram showing conditional mean have a name?

I'm trying to show the skill of a prediction $\hat{Y}$ relative to a true value $Y$ beyond simple statistics like MSE or $r^2$. I usually like looking at the scatter plot of $\hat{Y}$ vs $Y$ but in ...