This problem from ISI MStat 2016 is an application of the ideas of indicator and independent variables and covariance of two summative random variables.
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)
\( 2 = E(|X|) = E(X.1(X >0)) - E(X.1(X \leq 0)) = E(2X.1( X > 0 )) - E(X) = 2E(X.1( X > 0 ))\)
\( \Rightarrow E(X.1( X > 0 )) = 1 \overset{E(X) = 0}{\Rightarrow} Cov(X, 1( X > 0 )) = 1\).
Let's calculate the covariance of \(Y\) and \(Z\).
\( Cov(Y, Z) = \sum_{i,j = 1}^{n} Cov( X_i, 1(X_{j}>0))\)
\( = \sum_{i = 1}^{n} Cov( X_i, 1(X_{i}>0)) + \sum_{i,j = 1, i \neq j}^{n} Cov( X_i, 1(X_{j}>0)) \)
\( \overset{X_i \text{&} X_j \text{are independent}}{=} \sum_{i = 1}^{n} Cov( X_i, 1(X_{i}>0)) = \sum_{i = 1}^{n} 1 = n \).
This problem from ISI MStat 2016 is an application of the ideas of indicator and independent variables and covariance of two summative random variables.
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)
\( 2 = E(|X|) = E(X.1(X >0)) - E(X.1(X \leq 0)) = E(2X.1( X > 0 )) - E(X) = 2E(X.1( X > 0 ))\)
\( \Rightarrow E(X.1( X > 0 )) = 1 \overset{E(X) = 0}{\Rightarrow} Cov(X, 1( X > 0 )) = 1\).
Let's calculate the covariance of \(Y\) and \(Z\).
\( Cov(Y, Z) = \sum_{i,j = 1}^{n} Cov( X_i, 1(X_{j}>0))\)
\( = \sum_{i = 1}^{n} Cov( X_i, 1(X_{i}>0)) + \sum_{i,j = 1, i \neq j}^{n} Cov( X_i, 1(X_{j}>0)) \)
\( \overset{X_i \text{&} X_j \text{are independent}}{=} \sum_{i = 1}^{n} Cov( X_i, 1(X_{i}>0)) = \sum_{i = 1}^{n} 1 = n \).
