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Chapter 2: Problem 13

Let \(X_{1}\) and \(X_{2}\) be independent random variables with nonzero variances. Find the correlation coefficient of \(Y=X_{1} X_{2}\) and \(X_{1}\) in terms of the means and variances of \(X_{1}\) and \(X_{2}\).

Short Answer

Expert verified
The correlation coefficient of \(Y = X_{1}X_{2}\) and \(X_{1}\) in terms of the means and variances of \(X_{1}\) and \(X_{2}\) is given by: \(Correlation = \frac{E[X_{1}^2]E[X_{2}]-E[X_{1}]^2E[X_{2}]}{\sqrt{Var[X_{1}](Var[X_{1}]Var[X_{2}] + E[X_{1}]^2Var[X_{2}] + E[X_{2}]^2Var[X_{1}])}}\).

Step by step solution

01

Compute the Covariance

The covariance of two random variables \(X_{1}\) & \(Y\) is given by \(Cov[X_{1},Y] = E[(X_{1}-E[X_{1}])(Y-E[Y])]\). After substituting \(Y\) with \(X_{1}X_{2}\) and expanding, we derive to \(Cov[X_{1}, Y] = E[(X_{1}^2X_{2})]-E[X_{1}]E[X_{1}X_{2}]\). By applying linearity of expectation we get \(Cov[X_{1}, Y] = E[X_{1}^2]E[X_{2}]-E[X_{1}]^2E[X_{2}]\).
02

Compute Variance of X_{1} and Y

The variances of the random variables should be computed. We know that the variance of \(X_{1}\) already exists, \(Var[X_{1}]\), so we only need to compute for \(Y = X_{1}X_{2}\), which assuming independence leads to \(Var[Y] = Var[X_{1}]Var[X_{2}] + E[X_{1}]^2Var[X_{2}] + E[X_{2}]^2Var[X_{1}]\).
03

Calculate the Correlation Coefficient

The correlation coefficient of \(X_{1}\) & \(Y\) is given by \(Correlation = \frac{Cov[X_{1}, Y]}{\sqrt{Var[X_{1}]Var[Y]}}\). Substituting the expressions we got in Step 1 and Step 2 to compute the correlation, we get: \(Correlation = \frac{E[X_{1}^2]E[X_{2}]-E[X_{1}]^2E[X_{2}]}{\sqrt{Var[X_{1}](Var[X_{1}]Var[X_{2}] + E[X_{1}]^2Var[X_{2}] + E[X_{2}]^2Var[X_{1}])}}\).

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Most popular questions from this chapter

Let \(X_{1}\) and \(X_{2}\) have the joint pmf \(p\left(x_{1}, x_{2}\right)=x_{1} x_{2} / 36, x_{1}=1,2,3\) and \(x_{2}=1,2,3\), zero elsewhere. Find first the joint pmf of \(Y_{1}=X_{1} X_{2}\) and \(Y_{2}=X_{2}\), and then find the marginal pmf of \(Y_{1}\).

Suppose \(X_{1}\) and \(X_{2}\) are random variables of the discrete type which have the joint \(\operatorname{pmf} p\left(x_{1}, x_{2}\right)=\left(x_{1}+2 x_{2}\right) / 18,\left(x_{1}, x_{2}\right)=(1,1),(1,2),(2,1),(2,2)\), zero elsewhere. Determine the conditional mean and variance of \(X_{2}\), given \(X_{1}=x_{1}\), for \(x_{1}=1\) or 2. Also, compute \(E\left(3 X_{1}-2 X_{2}\right)\).

If the independent variables \(X_{1}\) and \(X_{2}\) have means \(\mu_{1}, \mu_{2}\) and variances \(\sigma_{1}^{2}, \sigma_{2}^{2}\), respectively, show that the mean and variance of the product \(Y=X_{1} X_{2}\) are \(\mu_{1} \mu_{2}\) and \(\sigma_{1}^{2} \sigma_{2}^{2}+\mu_{1}^{2} \sigma_{2}^{2}+\mu_{2}^{2} \sigma_{1}^{2}\), respectively.

If \(X\) has the pdf of \(f(x)=\frac{1}{4},-1

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