Question

Let \(X\) denote a random variable for which \(E\left[(X-a)^{2}\right]\) exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate distribution.

Short answer

An example of a degenerate distribution that would satisfy the given condition is a distribution where \(X\) equals a certain number \(a\) with a probability of 1. Thus, \(P(X=a) = 1\). In this scenario, \(E\left[(X-a)^{2}\right]\) equals 0, as all outcomes of \(X\) are \(a\).

Step-by-step solution

Understand the Identity

First, make sure you understand the given identity: \(E\left[(X-a)^{2}\right] = 0\). This means that the expected value of the square of the difference between the random variable \(X\) and a constant \(a\) is 0. This implies that, in all likelihood, the random variable \(X\) equals the constant \(a\).

Formulate The Degenerate Distribution

Use that information to formulate a degenerate distribution. If the expectation of the square of \(X-a\) is 0, it means that \(X=a\) almost surely. Hence, the random variable \(X\) can be considered as a constant value \(a\). In other words, the distribution of \(X\) is a degenerate distribution at point \(a\).

Identify The Example

Take an example of a degenerate distribution to fulfill the condition. Let the random variable \(X\) take on the constant value \(a\) with a probability of 1. This can be formalized as \(P(X=a) = 1\), and it is a valid discrete probability distribution that meets the given condition. The expected value \(E\left[(X-a)^{2}\right]\) will be 0, as \(X\) is \(a\) with total certainty.

Key concepts

Expected Value

The expected value, often represented as 'E', is a fundamental concept in probability and statistics that provides a measure of the center or average of a probability distribution. Think of it as the long-run average value of repetitions of the same experiment or process. For a discrete random variable, the expected value is calculated by summing the products of the possible values the variable can take and their corresponding probabilities.

When dealing with a degenerate distribution, where the random variable takes on a single value with certainty (probability 1), the expected value simplifies considerably. It is simply that value, since it's the only outcome possible every time we 'perform' the experiment. For example, if a random variable 'X' always equals 5, then the expected value of 'X' is also 5. In the context of the exercise, the expected value of \(E\left[(X-a)^{2}\right]\) being zero indicates that 'X' is always 'a' and therefore the expected value would also be 'a'.

Discrete Probability Distribution

A discrete probability distribution is a framework used to describe the likelihood of each of the different possible values of a discrete random variable. Discrete random variables are those that have countable outcomes, such as the number of heads when flipping a coin multiple times. This is opposed to continuous random variables, which can take an infinite number of values within a range, like heights of people.

In a discrete probability distribution, each possible value the random variable can assume is associated with a probability that ranges between 0 and 1. The sum of all such probabilities in a discrete distribution is always 1, since the probabilities represent the exhaustive set of outcomes. An example of a discrete probability distribution is the distribution of rolling a single six-sided die, where each face (1 through 6) has a probability of 1/6. In the exercise example, a degenerate distribution is a specific type of discrete distribution where the probability is 1 for the random variable being a specific value 'a' and 0 for all other possibilities, leading to a very predictable outcome.

Random Variable

A random variable is a numerical quantity whose value depends on the outcomes of a random phenomenon. It's like a placeholder that assigns numbers to outcomes of some random process. For instance, when you roll a dice, the outcome is random, and we can define a random variable 'X' to represent the result of that roll. In the case of a six-sided die, 'X' could be 1, 2, 3, 4, 5, or 6, depending on the roll, and each of these outcomes has an associated probability.

Random variables are broadly classified into two types: discrete and continuous. A discrete random variable has a countable number of possible values, while a continuous random variable can take on any value in a continuous range. As we consider the concept of a degenerate distribution, it is important to note that such a distribution concerns a discrete random variable that is essentially not 'random' at all, as its value is known with certainty. This idea is particularly useful in theoretical studies and certain practical applications where absolute certainty is implied.