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Chapter 4: Problem 106

Find the variance of the sum of 10 random variables if each has variance 5 and if each pair has correlation coefficient \(0.5\).

Short Answer

Expert verified
The variance of the sum of 10 random variables with the given characteristics is 275.

Step by step solution

01

Understand the Variables

We are told that there are 10 random variables, each with a variance of 5, and the correlation coefficient between each pair of variables is 0.5. The random variables are assumed to be pairwise, which means every pair of variables share the same correlation coefficient.
02

Use the Variance Sum Law

The variance of a sum of random variables is given by the Variance Sum Law: \[Var(X1+X2+...+Xn) = Var(X1) + Var(X2) + ... + Var(Xn) + 2*(Cov(X1,X2)+ Cov(X2,X3)+...+Cov(Xn,Xn-1))\n\] Where Xi are the random variables and Cov(Xi,Xj) represents the covariance between the variables Xi and Xj. Now we are required to use the given variables in this formula.
03

Calculate the Covariance from the Correlation

The correlation coefficient (r) is related to the covariance (Cov) by the formula: \[r = Cov / (SD1*SD2)\] Where SD1 and SD2 are the standard deviations of the variables, and for a variable X, \[SD = sqrt(Var(X))\] Because the random variables all have the same variance, their standard deviations will also be equal: \[SD1=SD2=sqrt(5)\], and the Covariance between two variables is therefore \[Cov = r * sqrt(Var(X1)) * sqrt(Var(X2)) = 0.5 * sqrt(5) * sqrt(5) = 2.5\].
04

Apply the Calculated Covariance to the Variance Sum Law

There are ten variables, but due to pairwise correlation, there are \[10*(10-1)/2=45\] covariance terms in the variance sum. Applied to the Variance Sum Law, we find \[Var(X1+X2+...+Xn) = 10*Var(Xi) + 2*45*Cov = 10*5 + 2*45*2.5 = 275.\]

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Most popular questions from this chapter

Let \(Y\) denote the median of a random sample of size \(n=2 k+1\), \(k\) a positive integer, from a distribution which is \(n\left(\mu, \sigma^{2}\right) .\) Prove that the graph of the p.d.f. of \(Y\) is symmetric with respect to the vertical axis through \(y=\mu\) and deduce that \(E(Y)=\mu\).

Find the p.d.f. of the sample variance \(S^{2}\), provided that the distribution from which the sample arises is \(n\left(\mu, \sigma^{2}\right)\).

Let \(X\) and \(Y\) be random variables with means \(\mu_{1}, \mu_{2}\); variances \(\sigma_{1}^{2}, \sigma_{2}^{2} ;\) and correlation coefficient \(\rho .\) Show that the correlation coefficient of \(W=a X+b, a>0\), and \(Z=c Y+d, c>0\), is \(\rho\)

If the sample size is \(n=2\), find the constant \(c\) so that \(S^{2}=\) \(c\left(X_{1}-X_{2}\right)^{2}\).

If \(X_{1}, X_{2}\) is a random sample from a distribution that is \(n(0,1)\), find the joint p.d.f. of \(Y_{1}=X_{1}^{2}+X_{2}^{2}\) and \(Y_{2}=X_{2}\) and the marginal p.d.f. of \(Y_{1}\). Hint. Note that the space of \(Y_{1}\) and \(Y_{2}\) is given by \(-\sqrt{y_{1}}
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