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Chapter 1: Problem 10

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
An example of a discrete distribution such that the expectation \(E[(X-a)^2]\) is zero is a degenerate distribution at an arbitrary point \(a\), where the random variable \(X\) equals \(a\) with probability 1. In this case, all possible outcomes are concentrated entirely on the value \(a\).

Step by step solution

01

Understand the expectation \(E[(X-a)^2]\)

The expression \(E[(X-a)^2]\) represents the expectation, or average, of the squared difference between the random variable \(X\) and a certain value \(a\). If its value is zero, that implies that the squared difference between \(X\) and \(a\) is always zero or simply \(X = a\) for all possible outcomes.
02

Define a random variable with corresponding degenerate distribution

A degenerate distribution is a probability distribution in which the possible outcomes are concentrated entirely on one value, hence it's always that value with probability 1. So, for an arbitrary real number \(a\), an example of a random variable could be \(X\) such that the probability \(P(X = a) = 1\). Thus, \(X\) follows a degenerate distribution at \(a\).
03

Verification

The expectation of \(X\) should equal to \(E[X] = a\), since X is always equal to \(a\) in the setting. The expectation \(E\[(X-a)^2\]\) would then simplify to \(E[(a-a)^2] = E[0] = 0\), which verifies our example.

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Most popular questions from this chapter

Let \(X\) have the pdf \(f(x)=3 x^{2}, 0

Suppose that \(p(x)=\frac{1}{5}, x=1,2,3,4,5\), zero elsewhere, is the pmf of the discrete-type random variable \(X\). Compute \(E(X)\) and \(E\left(X^{2}\right)\). Use these two results to find \(E\left[(X+2)^{2}\right]\) by writing \((X+2)^{2}=X^{2}+4 X+4\).

Find the moments of the distribution that has mgf \(M(t)=(1-t)^{-3}, t<1\). Hint: Find the Maclaurin series for \(M(t) .\)

For each of the following pdfs of \(X\), find \(P(|X|<1)\) and \(P\left(X^{2}<9\right)\). (a) \(f(x)=x^{2} / 18,-3

Let \(X\) be a random variable of the continuous type that has pdf \(f(x)\). If \(m\) is the unique median of the distribution of \(X\) and \(b\) is a real constant, show that $$ E(|X-b|)=E(|X-m|)+2 \int_{m}^{b}(b-x) f(x) d x $$ provided that the expectations exist. For what value of \(b\) is \(E(|X-b|)\) a minimum?
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