Question

Let \(X\) have the pmf \(p(x)=1 / k, x=1,2,3, \ldots, k\), zero elsewhere. Show that the \(\mathrm{mgf}\) is $$ M(t)=\left\\{\begin{array}{ll} \frac{e^{t}\left(1-e^{k t}\right)}{k\left(1-e^{t}\right)} & t \neq 0 \\ 1 & t=0 \end{array}\right. $$1.9.23. Let \(X\) have the pmf \(p(x)=1 / k, x=1,2,3, \ldots, k\), zero elsewhere. Show that the \(\mathrm{mgf}\) is $$ M(t)=\left\\{\begin{array}{ll} \frac{e^{t}\left(1-e^{k t}\right)}{k\left(1-e^{t}\right)} & t \neq 0 \\ 1 & t=0 \end{array}\right. $$

Short answer

The moment generating function is indeed \[M(t)=\left\{\begin{array}{ll}\frac{e^{t}(1-e^{kt})}{k(1-e^{t})} & t \neq 0 \ 1 & t=0 \end{array}\right.\] as proven above.

Step-by-step solution

Understand and write down the provided PMF

The given probability mass function of the discrete random variable \(X\) is \(p(x) = 1/k\) where x is an integer in the set \({1,2,3,...,k}\) and is zero otherwise.

Use the definition of MGF to write down its generic formula

The moment generating function \(M(t)\) of a random variable \(X\) is defined as the expected value of \(e^{tX}\), which can be calculated as \(M(t) = E(e^{tX}) = \sum (e^{tx})p(x)\) where the sum is over all possible values of \(X\).

Insert the provided PMF into the MGF formula

We insert the provided PMF into the formula of MGF to calculate it. Here, \(X\) can take values from 1 to \(k\), so we have \[M(t) = \sum_{x=1}^{k} e^{tx} * \frac{1}{k}\]

Simplify the resulting sum

We realize that we have a geometric series, which can be simplified to \[M(t) = \frac{e^{t} * (1-e^{kt})}{k*(1-e^{t})}\] for \(t != 0\). The sum of a geometric series with first term \(a\), ratio \(r\) and \(n\) terms is given by \(\frac{a*(1-r^n)}{(1-r)}\). Here, \(a = e^t\), \(r = e^t\) and \(n = k\). When \(t = 0\), the MGF is 1 by definition since \(e^{0}\) equals to 1.

Check the result

The resulting moment generating function matched the goal function, this showing the statement is true.

Key concepts

Probability Mass Function

Understanding the probability mass function (PMF) is key when working with discrete random variables. Essentially, the PMF is a function that provides the probabilities of a discrete random variable taking on specific values. In formal terms, if we have a discrete random variable, let's call it \( X \), then the PMF \( p(x) \) would tell us the probability that \( X \) equals some value \( x \), denoted as \( P(X = x) \).

For example, when a PMF is given by \( p(x) = 1/k \), for \( x \) in the set \( \{1,2,3,\ldots,k\} \), it is an indicator of a uniform distribution over these \( k \) integers. This means each value that \( X \) can take on is equally likely to occur, since each has the same probability \( 1/k \). This uniformity is critical in simplification processes, especially when dealing with the calculation of expected values and moment generating functions.

Discrete Random Variable

At the heart of our discussion is the concept of a discrete random variable. A discrete random variable \( X \) is a variable that can take on a countable number of values, often representing outcomes of an experiment or process. Unlike continuous random variables which can assume any value within certain bounds, discrete random variables have distinct and separate values.

When we say \( X \) is a discrete random variable with a PMF \( p(x) = 1/k \), we're describing a situation where \( X \) can assume one of \( k \) distinct outcomes. A classic example of a discrete random variable is the roll of a fair die, where the possible outcomes (1 through 6) are countable and each has the same probability of occurring.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio. In mathematical terms, a geometric series is given by \( a + ar + ar^2 + ar^3 + \ldots + ar^{n-1} \), where \( a \) denotes the first term, \( r \) is the common ratio, and \( n \) signifies the number of terms.

The sum of the geometric series can be found using the formula \( S_n = a \frac{1-r^n}{1-r} \), provided \( r \) is not equal to 1. The application of this series is widespread in mathematics and is particularly useful in finding the moment generating function (MGF) for certain probability distributions, as it simplifies the calculation by allowing us to combine terms mathematically.