# Questions tagged [conditional-expectation]

For every question related to the concept of conditional expectation of a random variable with respect to a $\sigma$-algebra. It should be used with the tag (probability-theory) or (probability), and other ones if needed.

2,798 questions
Filter by
Sorted by
Tagged with
22 views

17 views

25 views

### Determining convex function from all its expectations

Suppose I have a finite alphabet $\mathcal X$ and a convex function $h:\mathcal S(\mathcal X)\to \mathbb R$ from the simplex of probability distributions on $\mathcal X$ to the reals. Let $p_X$ be a ...
12 views

### Pareto analysis and conditional expectation

In a block of car insurance business you are considering there is a 50% chance that a claim will be made during the upcoming year. Once a claim is submitted the claim size follows Pareto with ...
38 views

19 views

### Defining Conditional Expectation through projections for non-square integrable rvs

This is from Asymptotic Statistics (van der Vaart) chapter 11 (problem 4). Let $X$ and $Y$ be random variable defined on the same probability space. Define $E[X|Y]$ as the measurable function $g_0$ ...
87 views

### Find the PDF of a random vector

So I have a question in probability theory that's driving me insane. I know it is easy but I can't seem to wrap my head around it. Assume we have $U_1\sim U[-1,1]$ and $U_2 \sim U[0,2]$ which are two ...
33 views

### Finding the distribution of a vector of random variables

I've encountered this question: Let (X,Y) be a jointly distributed normal vector. Find the distribution of (X, E(X|Y)) assuming Cov(X,Y) isn't singular. I'm unsure what to do - I tried using the joint ...
32 views

### Does $E[x_i\mid y_i] = 0 \implies \operatorname{cov}(x_i, y_i) = 0$?

$$E[x_i\mid y_i] = 0 \implies \operatorname{cov}(x_i, y_i) = 0\,?$$ I am wondering if the above statement holds true. The LHS is saying that given any value of $y_i$, the expected value of $x_i$ is ...
43 views

### Ratio of the expectation over the joint distribution and the expectation over the marginal distribution

I have the following expression and I am having trouble understanding it $$\frac{\int y f_{Y,X}(y,x) dy}{\int y f_Y(y) dy}$$ Where $f_{Y,X}(y,x)$ is the joint density function of random variables $X$ ...
116 views

45 views

### Proof that Expected Lifetime is longer than Remaining Lifetime if the Hazard Rate is increasing.

Let $X$ be a positive, continuous random variable. Denote the density of $X$ by $f(x)$ and its CDF by $F(x)$. Let $\bar{F}(x) = 1- F(x)$ be the survival function of $X$. Given that the Hazard Rate, \...
10 views

47 views

31 views

### Calculate $E(X\mid Y=y)$ and $E[YE(X\mid Y)]$ for $g(x,y)=6xy$

We know that (X,Y) have density $g(x,y)=6xy$ for $x,y \in T$ where $T=((x,y); 0 \leq x \leq 1, 0 \leq y\leq \sqrt x )$. The task is to calculate $E(X|Y=y)$ and $E[YE(X|Y)]$. Now I find joint density ...
40 views

59 views

### $\mathbb{E}(X|Y)$ and $\mathbb{P}(X \in A | Y)$ for r.v.s $X = x$ and $Y=\sin(\pi x)$ on $([0,1], \mathcal{B}_{[0,1]}, dx)$

Suppose we have r.v.s $X = x$ and $Y=\sin(\pi x)$ on $([0,1], \mathcal{B}_{[0,1]}, dx)$. what is $\sigma(Y)$? calculate $\mathbb{E}(X|Y)$ calculate $\mathbb{P}(X \in A | Y)$ for some borel set A from ...
35 views

### Question on expected waiting time calculation

A sequence of random variables, (not necessarily independent) are generated as follows. At each time $i \in [n]$, we toss a coin with head probability $\delta_i$ (these are random variables themselves)...
44 views

### Find $E[X\mid Y]$ for $y<x$

I'm having a dillemma regarding the integral of a density: Suppose: $$f_{X,Y}(x,y) = \frac{1}{2}xy\times\textbf{1}_{x\in[0,2]}\times\textbf{1}_{y\in[0,x]}$$ In order to find $E[X\mid Y]$ I know I want ...
17 views

### On the effect of random variables on the conditional expectations

How does the 'integrablity' of a random variable have impact on 'conditioning' on a random variable or vice-versa? When do we treat one random variable as a constant or just as a random variable in ...
69 views

Consider random variables $X$ and $Y$ with $\mathbb{E}[X^2] < \infty$. Show that if $g : \mathbb{R} \to \mathbb{R}$ is the function that minimizes $\mathbb{E}[(X -g(Y))^2]$, then $g(Y) = \mathbb{E}... 1answer 31 views ### if$X\mid Y$follows Bernoulli with parameter$g(Y)$then what is$E[X]$? The context is not important for the question but nevertheless here it is:$A$is the adjacency matrix of a random simple graph (A is symmetric with zero diagonal and with entries in$\{0,1\}$). The ... 1answer 48 views ### Find$\mathbb{P}_{\mathcal{F}}:2^{\{x_1,x_2,\dots\}}\times\Omega\rightarrow[0,1]$s.t.$\mathbb{P}_{\mathcal{F}}(\cdot,\omega)$is a prop.-measure Let$(\Omega,\Sigma,\mathbb{P})$be a probability space,$\mathcal{F}\subseteq\Sigma$a$\sigma$-Algbra and$X:\Omega\rightarrow\mathcal{X}$a random variable with a countable set$\mathcal{X}=\{x_1,...
Let $X_1,X_2$ be i.i.d. random variables with uniform law on $\{ 1, \dots N \}$. I want to compute $$E \left[ \max \{ X_1, X_2 \} \vert X_1 + X_2 \right].$$ How do I approach this? Do I need to ...