Question

Let \(Y\) denote the median and let \(\bar{X}\) denote the mean of a random sample of size \(n=2 k+1\) from a distribution that is \(N\left(\mu, \sigma^{2}\right)\). Compute \(E(Y \mid \bar{X}=\bar{x})\). Hint: See Exercise \(7.5 .4 .\)

Short answer

The conditional expectation of the median given the sample mean in a Normal distribution, \(E(Y \mid \bar{X}=\bar{x})\), is equal to the given sample mean, or \(E(Y \mid \bar{X}=\bar{x})=\bar{x}\).

Step-by-step solution

Understand the basic concept - Conditional Expectation

In probability theory, the conditional expectation of a random variable is the expected value of that variable given some other variable. Here, we want to calculate the expectation of median given that the mean is known.

Understanding Normal Distribution

Normal Distribution is parametrized by its mean (\(\mu\)) and standard deviation (\(\sigma\)). In this case, the distribution from which the sample is being selected is \(N(\mu, \sigma^{2})\). The sample size happens to be \(2k + 1\). Since the sample size is odd, the median will be an exact middle value of the arrayed data.

Understanding Median and Mean

The median is an average in the data distribution which separates the data into two halves, it can be assumed to be the center of the gaussian function. The mean \(\bar{X}\) is the average value of the dataset, and it is equal to \(\bar{x}\) according to the problem statement.

Calculation of the Expectation

Since our sampled distribution is symmetric (as it is a normal distribution), once we sort our sample and know the sample mean, the location of the median (middle of the distribution) is determined. Therefore, the conditional expected value of the median given the mean, is simply the known mean itself or \(E(Y \mid \bar{X}=\bar{x})=\bar{x}\). This is because, in a Normal distribution, mean equals median.

Key concepts

Probability Theory

Understanding probability theory is critical when tackling problems involving randomness and uncertainty, which is often the case in statistics. Probability theory forms the basis for making inferences about the characteristics of a population from a random sample. In essence, it helps us quantify the likelihood of certain outcomes and paves the way for concepts such as conditional expectation, which describes the average outcome of a random variable given that something is known about another variable.

For example, when flipping a fair coin, probability theory allows us to state that there is a 50% chance of landing on heads. Extending this, if we had information on preceding coin flips, conditional probability could guide us in adjusting our expectations for the next flip.

Normal Distribution

A normal distribution, often referred to as a bell curve, is a continuous probability distribution that is symmetric around its mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. Notably, in a normal distribution, the mean, median, and mode of the dataset are all equal.

A key feature is its two parameters, the mean \( \mu \) and standard deviation \( \sigma \) - the mean indicates the center of the distribution, while the standard deviation signifies the spread of the data around the mean. A fundamental property of the normal distribution is that it's completely characterized by these two parameters; this fact will be crucial when exploring the relationship between the sample mean and median.

Median and Mean

The median is the middle value in a dataset after it has been ordered from smallest to largest. If there’s an odd number of observations, the median is the middle number. If there’s an even number of observations, the median is the average of the two middle numbers. Unlike the mean, the median is not affected by extremely large or small values, making it a useful measure of central tendency when dealing with skewed distributions.

The mean, on the other hand, is the arithmetic average of a set of values or distribution and is susceptible to the influence of outliers. In a normal distribution, because of its symmetric nature, the mean aligns with the median, providing a central location around which the values are distributed.

Expectation Calculation

Calculating the expectation also known as the expected value, of a random variable, is finding the long-run average value that the variable takes on. In a probability distribution, the expected value is calculated by multiplying each possible outcome by its corresponding probability and summing all these products. It can be thought of as the 'center of mass' or the 'balance point' of the probability distribution.

In problems involving normal distributions and known conditions, such as a given sample mean \( \bar{X} = \bar{x} \), the expectation calculation simplifies. Since the normal distribution is symmetric and the mean equals the median, when we know the mean (or if it's given), it directly gives us the median value. Therefore, the conditional expectation of the median, in this case, is that same known mean value, which makes for a more straightforward computation.