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Chapter 4: Problem 1

Let \(\bar{X}_{n}\) denote the mean of a random sample of size \(n\) from a distribution that is \(N\left(\mu, \sigma^{2}\right) .\) Find the limiting distribution of \(\bar{X}_{n}\).

Short Answer

Expert verified
The limiting distribution of \(\bar{X}_{n}\) is a Normal distribution \`N(\mu, 0)\` as `n` approaches infinity.

Step by step solution

01

Identify Parameters of Given Distribution

The distribution from which the sample is taken, `N(\mu, \sigma^{2})`, is a normal distribution with mean `\mu` and variance `\sigma^{2}`.
02

Calculate the Mean and Variance of the Sample Mean

The mean of the sample mean `\bar{X}_{n}` can be calculated as the mean of the original distribution, which is `\mu`. The variance of `\bar{X}_{n}` is given by `\sigma^{2} / n`, where `n` is the size of the sample.
03

Apply Central Limit Theorem

The Central Limit Theorem tells us that, as `n` approaches infinity, the distribution of `\bar{X}_{n}` will approach a Normal distribution. Therefore, the limiting distribution of `\bar{X}_{n}` is also `N(\mu, \sigma^{2} / n)`, with `n 鈫� 鈭瀈.

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Most popular questions from this chapter

Let \(\bar{X}_{n}\) denote the mean of a random sample of size \(n\) from a Poisson distribution with parameter \(\mu=1\) (a) Show that the mgf of \(Y_{n}=\sqrt{n}\left(\bar{X}_{n}-\mu\right) / \sigma=\sqrt{n}\left(\bar{X}_{n}-1\right)\) is given by \(\exp \left[-t \sqrt{n}+n\left(e^{t / \sqrt{n}}-1\right)\right]\) (b) Investigate the limiting distribution of \(Y_{n}\) as \(n \rightarrow \infty\). Hint: Replace, by its MacLaurin's series, the expression \(e^{t / \sqrt{n}}\), which is in the exponent of the mgf of \(Y_{n}\).

Let \(Y_{n}\) denote the maximum of a random sample from a distribution of the continuous type that has cdf \(F(x)\) and pdf \(f(x)=F^{\prime}(x)\). Find the limiting distribution of \(Z_{n}=n\left[1-F\left(Y_{n}\right)\right]\)

Let \(Y_{1}

Let \(\bar{X}\) denote the mean of a random sample of size 100 from a distribution that is \(\chi^{2}(50)\). Compute an approximate value of \(P(49<\bar{X}<51)\).

For the last exercise, suppose that the sample is drawn from a \(N\left(\mu, \sigma^{2}\right)\) distribution. Recall that \((n-1) S^{2} / \sigma^{2}\) has a \(\chi^{2}(n-1)\) distribution. Use Theorem 3.3.1 to determine an unbiased estimator of \(\sigma\).
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