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 the required distribution could be a degenerate distribution where \(X=a\) with a probability of 1. This translates to \(P(X=a)=1\) and \(P(X=x)=0\) for all \(x\) not equal to \(a\).

Step-by-step solution

Understanding the expectation

The expectation \(E[(X-a)^2]\) can be interpreted as the average squared distance from \(X\) to \(a\). If this expectation is zero, it implies all of the distances are zero, meaning that \(X=a\) with probability 1.

Identify a suitable distribution

Accordingly, any distribution where \(X=a\) with probability 1 would satisfy this requirement. This is known as a degenerate distribution, which is a probability distribution in all of whose probability mass is concentrated at one point.

Specify the distribution

An example of a degenerate distribution would be a distribution where \(X=a\) with probability 1. In terms of a probability mass function, this could be written as \(P(X=a)=1\) and \(P(X=x)=0\) for all \(x\) not equal to \(a\).