Let $X_1$ and $X_2$ be i.i.d. random variables, and define $S_2 = X_1 + X_2$. Then $$\mathrm{E}[X_1|S_2] = \frac{X_1+X_2}2=\frac{S_2}2.$$ I don't understand why the conditional expectation is the average of the X's. Can anyone help me with that?
2026-04-24 19:21:23.1777058483
Conditional Expectation Value. Let $X_1$ and $X_2$ be i.i.d. random variables, and define $S_2$ = $X_1$ + $X_2$.
67 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in PROBABILITY-THEORY
- Is this a commonly known paradox?
- What's $P(A_1\cap A_2\cap A_3\cap A_4) $?
- Another application of the Central Limit Theorem
- proving Kochen-Stone lemma...
- Is there a contradiction in coin toss of expected / actual results?
- Sample each point with flipping coin, what is the average?
- Random variables coincide
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- Determine the marginal distributions of $(T_1, T_2)$
- Convergence in distribution of a discretized random variable and generated sigma-algebras
Related Questions in EXPECTED-VALUE
- Show that $\operatorname{Cov}(X,X^2)=0$ if X is a continuous random variable with symmetric distribution around the origin
- prove that $E(Y) = 0$ if $X$ is a random variable and $Y = x- E(x)$
- Limit of the expectation in Galton-Watson-process using a Martingale
- Determine if an Estimator is Biased (Unusual Expectation Expression)
- Why are negative constants removed from variance?
- How to find $\mathbb{E}(X\mid\mathbf{1}_{X<Y})$ where $X,Y$ are i.i.d exponential variables?
- $X_1,X_2,X_3 \sim^{\text{i.i.d}} R(0,1)$. Find $E(\frac{X_1+X_2}{X_1+X_2+X_3})$
- How to calculate the conditional mean of $E(X\mid X<Y)$?
- Let X be a geometric random variable, show that $E[X(X-1)...(X-r+1)] = \frac{r!(1-p)^r}{p^r}$
- Taylor expansion of expectation in financial modelling problem
Related Questions in CONDITIONAL-EXPECTATION
- Expectation involving bivariate standard normal distribution
- Show that $\mathbb{E}[Xg(Y)|Y] = g(Y) \mathbb{E}[X|Y]$
- How to prove that $E_P(\frac{dQ}{dP}|\mathcal{G})$ is not equal to $0$
- Inconsistent calculation for conditional expectation
- Obtaining expression for a conditional expectation
- $E\left(\xi\text{|}\xi\eta\right)$ with $\xi$ and $\eta$ iid random variables on $\left(\Omega, \mathscr{F}, P\right)$
- Martingale conditional expectation
- What is $\mathbb{E}[X\wedge Y|X]$, where $X,Y$ are independent and $\mathrm{Exp}(\lambda)$- distributed?
- $E[X|X>c]$ = $\frac{\phi(c)}{1-\Phi(c)}$ , given X is $N(0,1)$ , how to derive this?
- Simple example dependent variables but under some conditions independent
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Since $X_1=S_2-X_2$, we can rewrite the conditional expectation as $$\mathrm{E}[X_1|S_2] = \mathrm{E}[S_2-X_2|S_2] = S_2-\mathrm{E}[X_2|S_2].$$ Since $X_1$ and $X_2$ are i.i.d, $\mathrm{E}[X_1|S_2]=\mathrm{E}[X_2|S_2]$. Therefore, the equation above simplifies to $$2\mathrm{E}[X_1|S_2] = S_2.$$