Question

Let $X$ be a discrete uniform random variable assuming only the value $c$ ( $X$ is thus a constant random variable). Show that for each positive integer $n$, $$ E\left[X^n\right]=E[X]^n=c^n . $$

Name: Problem 40, Chapter 3: Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Uploaded: 2021-08-21T18:13:30-08:00
Duration: 1 min 23 s
Channel: Yingtai Xiao
Description: Let X be a discrete uniform random variable assuming only the value c ( X is thus a constant random variable). Show that for each positive integer n, E[X^n]=E[X]^n=c^n .

   Let $X$ be a discrete uniform random variable assuming only the value $c$ ( $X$ is thus a constant random variable). Show that for each positive integer $n$,
$$
E\left[X^n\right]=E[X]^n=c^n .
$$

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 3, Problem 40

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Solved by Expert Yingtai Xiao

08/21/2021

Step 1

For a discrete random variable $X$, the expectation $E[X]$ is defined as: $$ E[X] = \sum_x x \cdot P(X = x) $$ where the sum is taken over all possible values $x$ that $X$ can take, and $P(X = x)$ is the probability that $X$ equals $x$. Show more…

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Let $X$ be a discrete uniform random variable assuming only the value $c$ ( $X$ is thus a constant random variable). Show that for each positive integer $n$, $$ E\left[X^n\right]=E[X]^n=c^n . $$

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Key Concepts

Constant Random Variable

A constant random variable is one that takes on a single value with probability one. In this case, since the variable always equals c, its distribution is degenerate, meaning there is no variability and all probabilistic analysis simplifies as the outcome is certain.

Expectation

The expectation or mean of a random variable is the weighted average of all possible values it can take. For a constant random variable, the expectation is exactly the constant itself because the variable only assumes that one value, thereby making calculations straightforward.

Moments

Moments are quantitative measures related to the shape of a random variable’s probability distribution, defined as the expected values of powers of the variable. When dealing with a constant random variable, the nth moment is simply c raised to the nth power, since every instance of the variable is c and this property is directly applied to all powers.

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