# Discover the Covariance | ISI MStat 2016 Problem 6

This problem from ISI MStat 2016 is an application of the ideas of indicator and independent variables and covariance of two summative random variables.

## Problem- Covariance Problem

Let $X_{1}, \ldots, X_{n}$ ~ $X$ be i.i.d. random variables from a continuous distribution whose density is symmetric around 0. Suppose $E\left(\left|X\right|\right)=2$ . Define $Y=\sum_{i=1}^{n} X_{i} \quad \text { and } \quad Z=\sum_{i=1}^{n} 1\left(X_{i}>0\right)$.
Calculate the covariance between $Y$ and $Z$.

This problem is from ISI MStat 2016 (Problem #6)

### Prerequisites

1. X has Symmetric Distribution around 0 $\Rightarrow E(X) = 0$.
2. $|X| = X.1( X > 0 ) - X.1( X \leq 0 ) = 2X.1( X > 0 ) - X$, where $X$ is a random variable.
3. $X_i$ and $X_j$ are independent $\Rightarrow$ $g( X_i)$ and $f(X_j)$ are independent.
4. $A$ and $B$ are independent $\Rightarrow Cov(A,B) = 0$.

## Solution

$2 = E(|X|) = E(X.1(X >0)) - E(X.1(X \leq 0)) = E(2X.1( X > 0 )) - E(X) = 2E(X.1( X > 0 ))$

$\Rightarrow E(X.1( X > 0 )) = 1 \overset{E(X) = 0}{\Rightarrow} Cov(X, 1( X > 0 )) = 1$.

Let's calculate the covariance of $Y$ and $Z$.

$Cov(Y, Z) = \sum_{i,j = 1}^{n} Cov( X_i, 1(X_{j}>0))$

$= \sum_{i = 1}^{n} Cov( X_i, 1(X_{i}>0)) + \sum_{i,j = 1, i \neq j}^{n} Cov( X_i, 1(X_{j}>0))$

$\overset{X_i \text{&} X_j \text{are independent}}{=} \sum_{i = 1}^{n} Cov( X_i, 1(X_{i}>0)) = \sum_{i = 1}^{n} 1 = n$.

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