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I.S.I. and C.M.I. Entrance ISI M.Stat PSB Probability

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.

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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