(a) Is there a discrete distribution with support 1,2,3, …, such that the value of the PMF at n

step 1 So we're given the probability is equal to 1 over n, and x could be any number from 1 to n. And

(a) Is there a discrete distribution with support 1,2,3, …,...

(a) Is there a discrete distribution with support $1,2,3, \ldots$, such that the value of the PMF at $n$ is proportional to $1 / n$ ? Hint: See the math appendix for a review of some facts about series. (b) Is there a discrete distribution with support $1,2,3, \ldots$, such that the value of the PMF at $n$ is proportional to $1 / n^{2}$ ?

(a) Is there a discrete distribution with support $1,2,3, \ldots$, such that the value of the PMF at $n$ is proportional to $1 / n$ ?
Hint: See the math appendix for a review of some facts about series.
(b) Is there a discrete distribution with support $1,2,3, \ldots$, such that the value of the PMF at $n$ is proportional to $1 / n^{2}$ ?

Key Concepts

Discrete Probability Distribution

A discrete probability distribution is a function that assigns probabilities to each outcome in a countable sample space, ensuring that the sum of the probabilities is one. When constructing a distribution where the probability mass function (PMF) is proportional to a given function, it is essential that this function can be normalized by finding a constant such that the cumulative sum over all outcomes converges to one.

Normalization of a Probability Mass Function (PMF)

Normalization is the process of scaling a positive function so that the total probability over the support equals one. In cases where the unnormalized function is given as a proportionality, one must compute the sum over the entire support. If the sum converges, a normalizing constant exists; if it diverges, the function cannot be normalized into a valid PMF.

Convergence and Divergence of Series

Determining whether a function can serve as a PMF often involves checking the convergence of the infinite series summing the unnormalized probabilities over the support. A convergent series results in a finite sum that can be inverted to normalize the distribution, while a divergent series indicates that no finite constant can standardize the series to sum to one.

p-Series and Harmonic Series

A p-series is an infinite series of the form ?1/n^p. Its convergence depends critically on the exponent p. The harmonic series, where p = 1, is a well-known example that diverges, meaning that any PMF proportional to 1/n cannot be normalized over the natural numbers. In contrast, for p > 1, such as p = 2 for the series ?1/n^2, the series converges, making it possible to choose a normalization constant so that the resulting function is a valid probability mass function.

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