Let X,Y are independent random variables with equal distrbution (identically distributed). Show, that E(X| X + Y ) = E(Y |X + Y ) = (X + Y) / 2
2026-03-31 03:28:07.1774927687
Show the independence between two i.i.d random variables
37 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 MULTIVARIABLE-CALCULUS
- Equality of Mixed Partial Derivatives - Simple proof is Confusing
- $\iint_{S} F.\eta dA$ where $F = [3x^2 , y^2 , 0]$ and $S : r(u,v) = [u,v,2u+3v]$
- Proving the differentiability of the following function of two variables
- optimization with strict inequality of variables
- How to find the unit tangent vector of a curve in R^3
- Prove all tangent plane to the cone $x^2+y^2=z^2$ goes through the origin
- Holding intermediate variables constant in partial derivative chain rule
- Find the directional derivative in the point $p$ in the direction $\vec{pp'}$
- Check if $\phi$ is convex
- Define in which points function is continuous
Related Questions in STATISTICAL-INFERENCE
- co-variance matrix of discrete multivariate random variable
- Question on completeness of sufficient statistic.
- Probability of tossing marbles,covariance
- Estimate the square root of the success probability of a Binomial Distribution.
- A consistent estimator for theta is?
- Using averages to measure the dispersion of data
- Confidence when inferring p in a binomial distribution
- A problem on Maximum likelihood estimator of $\theta$
- Derive unbiased estimator for $\theta$ when $X_i\sim f(x\mid\theta)=\frac{2x}{\theta^2}\mathbb{1}_{(0,\theta)}(x)$
- Show that $\max(X_1,\ldots,X_n)$ is a sufficient statistic.
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
Related Questions in CONDITIONAL-PROBABILITY
- Given $X$ Poisson, and $f_{Y}(y\mid X = x)$, find $\mathbb{E}[X\mid Y]$
- Finding the conditional probability given the joint probability density function
- Easy conditional probability problem
- Conditional probability where the conditioning variable is continuous
- probability that the machine has its 3rd malfunction on the 5th day, given that the machine has not had three malfunctions in the first three days.
- Sum of conditional probabilities equals 1?
- Prove or disprove: If $X | U$ is independent of $Y | V$, then $E[XY|U,V] = E[X|U] \cdot E[Y|V]$.
- Conditional probability and binomial distribution
- Intuition behind conditional probabilty: $P(A|B)=P(B\cap A)/P(B)$
- Transition Probabilities in Discrete Time Markov Chain
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?
By symmetry, we have $E(X\mid X + Y) = E(Y \mid X+Y)$ (good exercise to formally justify this with the definition of conditional expectation).
Hence $X+Y = E(X+Y\mid X+Y) = E(X\mid X+Y) + E(Y\mid X+Y) = 2 E(X\mid X+Y)$
Thus $E(X \mid X+Y) = (X+Y)/2= E(Y \mid X+Y)$.