Deriving the Moment Generating Function of a Discrete Random Variable with Special Probability Density Function

In this article, we delve into the methodology of finding the moment generating function (MGF) of a discrete random variable X. Given that the probability density function (PDF) of X is (f_X frac{6}{pi x^2}) for (x in mathbb{N}), the MGF is defined as the expected value of (e^{tX}).

Understanding the Conditions and Calculations

The moment generating function (MGF) is a key concept in probability theory and statistics that generates the moments of a random variable. The MGF of a discrete random variable X is defined by the formula:

[ M(t) E[e^{tX}] ]

Given the specific PDF for X:

[ f_X(x) frac{6}{pi x^2} text{ for } x in mathbb{N} ]

The MGF can be derived as follows:

Step 1: Calculate the MGF using the definition

By definition, we need to find:

[ M(t) sum_{x1}^{infty} e^{tx} f_X(x) ]

Substitute the given PDF:

[ M(t) sum_{n1}^{infty} e^{tn} cdot frac{6}{pi n^2} ]

Step 2: Simplify the Summation

We can simplify the summation by factoring out constants:

[ M(t) frac{6}{pi^2} sum_{n1}^{infty} frac{e^{tn}}{n^2} ]

The series (sum_{n1}^{infty} frac{e^{tn}}{n^2}) is a special series that can be expressed as the polylogarithm function of order 2, denoted as (text{Li}_2(e^t)). Therefore:

[ M(t) frac{6}{pi^2} text{Li}_2(e^t) ]

Step 3: Analyze the Validity of the MGF

In this specific case, we need to consider the conditions under which the MGF is defined. The given moment generating function is defined for:

[ t For (t [ M(t) frac{6}{pi^2} text{Li}_2(e^t) ]

This function represents the moment generating function of the random variable X. However, for (t geq 0), the series may diverge, indicating that the random variable X does not have finite moments of any order. In such cases, we cannot use the MGF to derive moments of X.

Characteristics of the Random Variable X

The random variable X described in the problem does not possess any finite moments. This characteristic is reflected in the fact that:

[ E(X^k) infty text{ for any positive integer } k ]

This means that the moment generating function is undefined for (t geq 0) because the expected value (E[e^{tX}]) is undefined in this range.

The Characteristic Function of X

While the moment generating function is not useful for generating moments, the characteristic function of X, defined by:

[ varphi(t) E[e^{itX}] ]

can still be calculated. For our random variable X:

[ varphi(t) text{Li}_2(e^{it}) ]

Conclusion

In summary, we have derived the moment generating function of a discrete random variable X with a specific probability density function. The MGF is given by:

[ M(t) frac{6}{pi^2} text{Li}_2(e^t) text{ for } t This expression provides valuable information about the behavior of X while highlighting the limitations of the MGF when dealing with random variables that have no finite moments.