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Chapter 2: Problem 16

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 the 10 variables is 162.5.

Step by step solution

01

Understand Objectives and Definitions

You are asked to find the variance of the sum of 10 random variables. It's given that each of these variables has a variance of 5 and pair-wise correlation of 0.5. Variance is a measure of how much values in the set differ from the mean, and correlation is a measure of how much two random variables change together.
02

Define Variables and Apply Formula

Define each random variable as \(X_i\) for \(i=1\) to 10. Thus, we want to find the variance of the sum of this set of variables, \( S = X_1 + X_2 + ... + X_{10}\). The variance can be calculated by using the formula: \[Var(S) = Var(X_1 + X_2 + ... + X_{10}) = \sum_{i=1}^{10} Var(X_i) + \sum_{i\neq j} Cov(X_i, X_j)\]
03

Calculate Variance

The variance for each variable is given as 5, and there are 10 variables. So, the total variance \(\sum_{i=1}^{10} Var(X_i)= 5 * 10 = 50 \) .
04

Calculate Covariance

Covariance can be calculated using the given correlation coefficient and the product of standard deviations of the two variables. It is given that the correlation coefficient for each pair is 0.5. So, \(Cov(X_i, X_j) = \rho_i_j \sigma_i \sigma_j\), which equals \(0.5 * \sqrt{5} * \sqrt{5}\) which equals \(0.5 * 5 = 2.5\). There are \(C(10, 2) = 45\) pairs of 2 variables in 10, so total covariance \(\sum_{i\neq j} Cov(X_i, X_j) = 45 * 2.5 = 112.5 \) .
05

Final Calculation

Finally, plug the sums of the variance and covariance into the formula for the total variance of the sum of the variables. So, total variance \(Var(S) = 50 + 112.5 = 162.5 \).

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Most popular questions from this chapter

Let \(X_{1}, X_{2}, X_{3}\) be iid, each with the distribution having pdf \(f(x)=e^{-x}, 0<\) \(x<\infty\), zero elsewhere. Show that $$ Y_{1}=\frac{X_{1}}{X_{1}+X_{2}}, \quad Y_{2}=\frac{X_{1}+X_{2}}{X_{1}+X_{2}+X_{3}}, \quad Y_{3}=X_{1}+X_{2}+X_{3} $$ are mutually independent.

Let \(X_{1}, X_{2}, X_{3}\) be iid with common pdf \(f(x)=e^{-x}, x>0,0\) elsewhere. Find the joint pdf of \(Y_{1}=X_{1} / X_{2}, Y_{2}=X_{3} /\left(X_{1}+X_{2}\right)\), and \(Y_{3}=X_{1}+X_{2}\). Are \(Y_{1}, Y_{2}, Y_{3}\) mutually independent?

Let \(X_{1}\) and \(X_{2}\) have the joint \(\operatorname{pdf} h\left(x_{1}, x_{2}\right)=8 x_{1} x_{2}, 0

Find the mean and variance of the sum \(Y=\sum_{i=1}^{5} X_{i}\), where \(X_{1}, \ldots, X_{5}\) are iid, having pdf \(f(x)=6 x(1-x), 0

If the correlation coefficient \(\rho\) of \(X\) and \(Y\) exists, show that \(-1 \leq \rho \leq 1\). Hint: Consider the discriminant of the nonnegative quadratic function $$ h(v)=E\left\\{\left[\left(X-\mu_{1}\right)+v\left(Y-\mu_{2}\right)\right]^{2}\right\\} $$ where \(v\) is real and is not a function of \(X\) nor of \(Y\).
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