# Expectation value of the sum of two random variables

by jg370
Tags: expectation, random, variables
 P: 18 1. The problem statement, all variables and given/known data The expectation value of the sum of two random variables is given as: $$\langle x + y \rangle = \langle x \rangle + \langel y \rangle$$ My textbook provides the following derivation of this relationship. Suppose that we have two random variables, x and y. Let $$p_{ij}$$ be the probability that our measurement returns $$x_{i}$$ for the value of x and $$y_{j}$$ for the value of y. Then the expectation value of the sum of $$x+y$$ is: $$\langle x + y \rangle = \sum\limits_{ij} p_{ij} (x_i + y_j) =\sum\limits_{ij} p_{ij} x_i + \sum\limits_{ij} p_{ij} x_j$$ Then I am given the following statement: But $$\sum\limits_j p_{ij} = p_i$$ is the probability that we measure $$x_i$$ regardless of what we measure for y, so it must be equal to $$p_i$$. Similarly, $$\sum\limits_i p_{ij} = p_j$$, is the probability of measuing $$y_i$$ irrespective of what we get for $$x_i$$. 2. Relevant equations The difficulty I have with this statement is that I do no see how $$\sum\limits_j p_{ij}$$ can be equal to $$p_i$$. 3. The attempt at a solution Summing over j, we should have $$(p_{i1} + p_{i2},+ .... p_{in})$$. Now, is this equal to $$p_i$$. And similarly how can $$\sum\limits_i p_{ij}$$ can be equal to $$p_j$$ I am hopefull that someone can clear this up for me. Thank you for your kind assitance. jg370[/quote] 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
 HW Helper P: 3,309 so you have 2 discrete random variables X & Y, with a joint distribution, pij $$p_{ij} = P(X=x_i, Y = y_j)$$ the expectation is given by: $$= \sum_{ij} p_{ij} x_i y_j =$$ By definition, the marginal probabilities are $$P(X=x_i) = \sum_{j} p_{ij} = p_i$$ $$P(Y=y_j) = \sum_{i} p_{ij} = p_j$$ If the variables are independent then you have the further conidtion that $$p_{ij} = P(X=x_i, Y = y_j) = P(X=x_i)P(Y = y_j) = p_i p_j$$

 Related Discussions Precalculus Mathematics Homework 1 Calculus & Beyond Homework 3 Calculus & Beyond Homework 0 Set Theory, Logic, Probability, Statistics 5 Calculus & Beyond Homework 0