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Chapter 1: Problem 93

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

Expert verified
A distribution that satisfies the given condition is a degenerate distribution, where the value of the random variable is always \(a\).

Step by step solution

01

Understand the problem

The exercise is asking for a distribution which meets the criteria \(E[(X-a)^2] = 0\). This expression equals zero when every value of the random variable equals to \(a\). So, it is necessary to construct a distribution in such a way.
02

Define the degenerate distribution

In statistics, a degenerate distribution is a probability distribution concentrated at a single value. This is also known as a 'delta function'. By the definition, we could say that \(X\) follows a degenerate distribution at \(a\) if the probability \(P(X=a) = 1\) and \(P(X=x) = 0\) for all \(x 鈮� a\)..
03

Calculate \(E(X-a)^2\)

The expectation \(E\) of \((X-a)^2\) is the sum over all possible values of \((X-a)^2\) times their respective probabilities. In a degenerate distribution at \(a\), all other values except \(a\) have probability zero. Hence, \(E[(X-a)^2] = P(X=a) \cdot (X-a)^2 = 1 \cdot 0 = 0\).

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