Let Y1<Y2 denote the order statistics of a random sample of...

Let $Y_{1}<Y_{2}$ denote the order statistics of a random sample of size 2 from a distribution that is $N\left(\mu, \sigma^{2}\right)$, where $\sigma^{2}$ is known. (a) Show that $P\left(Y_{1}<\mu<Y_{2}\right)=\frac{1}{2}$ and compute the expected value of the random length $Y_{2}-Y_{1}$. (b) If $\bar{X}$ is the mean of this sample, find the constant $c$ that solves the equation $P(\bar{X}-c \sigma<\mu<\bar{X}+c \sigma)=\frac{1}{2}$, and compare the length of this random interval with the expected value of that of part (a).

Let $Y_{1}<Y_{2}$ denote the order statistics of a random sample of size 2 from a distribution that is $N\left(\mu, \sigma^{2}\right)$, where $\sigma^{2}$ is known.
(a) Show that $P\left(Y_{1}<\mu<Y_{2}\right)=\frac{1}{2}$ and compute the expected value of the random length $Y_{2}-Y_{1}$.
(b) If $\bar{X}$ is the mean of this sample, find the constant $c$ that solves the equation $P(\bar{X}-c \sigma<\mu<\bar{X}+c \sigma)=\frac{1}{2}$, and compare the length of this random interval with the expected value of that of part (a).

Key Concepts

Coverage Probability and Interval Estimation

Coverage probability refers to the likelihood that a confidence interval contains the true value of the parameter being estimated. In the context of the problem, determining the constant 'c' involves setting the interval around the sample mean such that it has a specified probability (½) of containing the true mean. This requires understanding the distribution of the estimator (in this case, the sample mean) and performing the appropriate standardization.

Order Statistics

Order statistics refer to the sorted values from a sample. In statistical theory, they are used to analyze the behavior of the smallest, largest, or other ordered sample points. Understanding order statistics is crucial when you need to compute probabilities related to the minimum and maximum values of a sample, such as the probability that the true mean lies between them or the expected difference (range) between the smallest and largest observations.

Symmetry of the Normal Distribution

The normal distribution is symmetric about its mean, which greatly simplifies many probability calculations. This symmetry implies that events positioned symmetrically around the mean have equal probabilities. In problems involving order statistics or sample means from a normal distribution, this symmetry is used to conclude, for example, that the probability of the true mean lying between the smallest and largest sample values is one half.

Distribution of the Sample Mean

When sampling from a normal distribution, the sample mean is also normally distributed. Its mean equals the population mean and its variance is reduced by the sample size. For a sample of size 2, the variance of the sample mean is ?²/2. This property is essential when constructing confidence intervals or determining the probability that an interval centered at the sample mean captures the true population mean.

Expected Difference (Range) Between Order Statistics

The expected difference between the largest and smallest values of a sample, known as the range, is a measure of variability inherent in the data gathered from a population. Calculating the expected range involves the distribution properties of order statistics. This concept serves to compare the inherent scatter in the data with the width of intervals constructed from the sample mean.

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