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# 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 $$Z$$.

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

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

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