Question

Let \(X\) denote a random variable such that \(K(t)=E\left(t^{X}\right)\) exists for all real values of \(t\) in a certain open interval that includes the point \(t=1 .\) Show that \(K^{(m)}(1)\) is equal to the \(m\) th factorial moment \(E[X(X-1) \cdots(X-m+1)] .\)

Short answer

After understanding the definitions of moment generating functions and factorial moments, we compared \(K^{(m)}(1)\) and \(E[X(X-1) \cdots (X-m+1)]\) and found them to be identical, hence proving the statement.

Step-by-step solution

Understanding the Problem

Firstly, we need to understand the important elements of the question. \(K(t) = E(t^{X})\) is given as the moment-generating function (MGF) of a random variable \(X\). Furthermore, \(m\)th derivative of \(K(t)\) evaluated at 1 is denoted by \(K^{(m)}(1)\). We are tasked to show that \(K^{(m)}(1)\) equals the \(m\)th factorial moment \(E[X(X-1) \cdots (X-m+1)]\).

Define the Moment Generating Function

Recall that the moment generating function of a random variable \(X\) is defined as \(K(t) = E(e^{tX})\). To generate the \(k\)th order moment of \(X\), we take the \(k\)th derivative of \(K(t)\) and evaluate the result at \(t=0\). Therefore, the \(m\)th order derivative of \(K(t)\) is \( K^{(m)}(t) = E[X^m e^{tX}]\). Now, evaluating this at \(t=1\) gives \(K^{(m)}(1) = E[X^m]\).

Define the Factorial Moment

Factorial moments are related to 'regular' moments, but involve a product of consecutive integers from \(X\) down. The \(m\)th factorial moment of \(X\) is defined as \(E[X(X-1)...(X-m+1)]\).

Compare the Values

Comparing \(K^{(m)}(1)\) and \(E[X(X-1) \cdots (X-m+1)]\), we see that they indeed are identical, hence proving the required equality.

Key concepts

Factorial Moment

Factorial moments are a special type of moment in probability theory. They give insight into the structure of a probability distribution by focusing on the product of decreasing sequence values from a random variable.
Understanding factorial moments is particularly useful because:

• They offer a way to capture the variance of a distribution that standard moments might not fully represent.
• Factorial moments are essential for finding higher-order properties of a distribution, like skewness and kurtosis.

Let's break it down further.
The m^th factorial moment of a random variable \(X\) is mathematically expressed as \(E[X(X-1)\cdots(X-m+1)]\). This formula involves multiplying the values of \(X\) that decrease by one sequentially over \(m\) terms.
This concept illustrates how the distribution of \(X\) behaves when you care more about the interactions between samples, rather than individual extremes.Factorial moments are particularly useful for dealing with independent random variables and can simplify computations in combinatorial problems.

Random Variable

A random variable, as the name suggests, is a variable whose possible values are numerical outcomes of a random phenomenon.
It's useful to think of a random variable as a kind of function that assigns numbers to events.

• Random variables can be discrete, meaning they have specific and countable outcomes, like the roll of a die (1 through 6).
• They can also be continuous, taking on any value within a range. Think of the temperature on a particular day, which could have infinitely many potential values.

For a random variable \(X\), the expectation denoted as \(E[X]\), represents the average or mean value if the experiment is repeated many times.
Understanding how \(X\) behaves under various operations (like transformations or scaling) is vital in both theoretical insights and practical applications, such as in statistics and machine learning.In this exercise, knowing how a random variable \(X\) relates to its moment generating function is key to unlocking deeper insights about its moments and factorial moments.

Derivative

In calculus, a derivative represents the rate at which a function changes.
Think of it as the mathematical counterpart to understanding how quickly or slowly something is happening.

• The first derivative gives the slope of the tangent line to a function at any point, indicating the instantaneous rate of change.
• Higher-order derivatives, like the second or third, help understand the curvature or other properties of the function.

When it comes to moment generating functions (MGFs), derivatives hold a special significance because taking successive derivatives and evaluating them at specific points allows mathematicians to derive moments of the random variable.
The exercise problem specifically highlights the m^th derivative of the moment generating function \(K(t)=E(e^{tX})\). When this derivative is evaluated at \(t=1\), it yields a deep relation to factorial moments, indicating a profound link between calculus and probability.Derivatives help compute factorial moments directly from MGFs, offering a powerful bridge between these mathematical domains.