/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Let \(Z_{n}\) be \(\chi^{2}(n)\)... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 8

Let \(Z_{n}\) be \(\chi^{2}(n)\) and let \(W_{n}=Z_{n} / n^{2}\). Find the limiting distribution of \(W_{n}\).

Short Answer

Expert verified
The limiting distribution of \(W_{n}\) is \(0\).

Step by step solution

01

Identify the Distribution of \(Z_{n}\)

First, identify that the random variable \(Z_{n}\) has a chi-square distribution with \(n\) degrees of freedom. A chi-square distribution with \(n\) degrees of freedom has a probability density function given by: \[f(x) = \frac{1}{{2^{n/2}}{\Gamma(n/2)}} x^{n/2 - 1} e^{-x/2} \], for \(x > 0\). \(Z_n\) also has a mean of \(n\) and variance \(2n\).
02

Characterize the sequence \(W_{n}\)

The next step is to convert the sequence \(Z_{n}\) into \(W_{n}\). We can achieve this by dividing \(Z_{n}\) by \(n^2\), which is equivalent to scaling the \(Z_{n}\) random variable. Consequently, the new random variable \(W_{n}\) satisfies \(W_{n} = Z_{n}/n^2\).
03

Obtain the limiting distribution

The limiting distribution of the sequence \(W_{n}\) will be a distribution such that for any continuous function \(g\), the sequence of expected values \(E(g(W_n))\) converges to the expected value \(E(g(W))\), where \(W\) is a random variable with the limiting distribution. For the given exercise, by strong law of large numbers, which states that the sample average converges almost surely to the expected value, we have the limiting distribution as \(0\), since as \(n\) approaches infinity, \(W_{n}\) approaches \(0\).

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Most popular questions from this chapter

Let \(X\) and \(Y\) have the parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho .\) Show that the correlation coefficient of \(X\) and \(\left[Y-\rho\left(\sigma_{2} / \sigma_{1}\right) X\right]\) is zero.

Let \(X_{1}\) and \(X_{2}\) have a bivariate normal distribution with parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho .\) Compute the means, the variances, and the correlation coefficient of \(Y_{1}=\exp \left(X_{1}\right)\) and \(Y_{2}=\exp \left(X_{2}\right)\) Hint: Various moments of \(Y_{1}\) and \(Y_{2}\) can be found by assigning appropriate values to \(t_{1}\) and \(t_{2}\) in \(E\left[\exp \left(t_{1} X_{1}+t_{2} X_{2}\right)\right]\).

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.

Let \(X_{1}\) and \(X_{2}\) have a joint distribution with parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho\). Find the correlation coefficient of the linear functions of \(Y=a_{1} X_{1}+a_{2} X_{2}\) and \(Z=b_{1} X_{1}+b_{2} X_{2}\) in terms of the real constants \(a_{1}, a_{2}, b_{1}, b_{2}\), and the parameters of the distribution.

Let the random variable \(Z_{n}\) have a Poisson distribution with parameter \(\mu=n\). Show that the limiting distribution of the random variable \(Y_{n}=\left(Z_{n}-n\right) / \sqrt{n}\) is normal with mean zero and variance \(1 .\)
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