Problem 12 - A Computational Introduction to Number Theory and Algebra

Question

Suppose \(X\) and \(X^{\prime}\) are random variables that take values in a set \(S\) and that have essentially the same distribution. Show that if \(f: S \rightarrow T\) is a function, then \(f(X)\) and \(f\left(X^{\prime}\right)\) have essentially the same distribution.

Step-by-Step Solution

Answer

Question: Show that if random variables \(X\) and \(X^{\prime}\) have essentially the same distribution, then the random variables \(f(X)\) and \(f\left(X^{\prime}\right)\) have essentially the same distribution for a function \(f: S \rightarrow T\). Answer: If \(X\) and \(X^{\prime}\) have essentially the same distribution, their PMFs (or PDFs) are equal. Therefore, the PMFs (or PDFs) of \(f(X)\) and \(f\left(X^{\prime}\right)\), denoted as \(Y\) and \(Y^{\prime}\) respectively, are also equal: \(p_Y(t) = p_{Y^{\prime}}(t)\). Thus, \(f(X)\) and \(f\left(X^{\prime}\right)\) have essentially the same distribution.

Step 1: Rewrite the random variables \(f(X)\) and \(f(X^{\prime})\)

Let \(Y = f(X)\) and \(Y^{\prime} = f(X^{\prime})\). We have two new random variables, \(Y\) and \(Y^{\prime}\), and we need to determine their PMFs or PDFs.

Step 2: Compare the PMFs or PDFs

Since \(X\) and \(X^{\prime}\) have essentially the same distribution, their PMFs (or PDFs) are equal. Let's denote the PMF of \(X\) as \(p_X(s)\), where \(s \in S\). We can write the PMF of \(Y\) as: \(p_Y(t) = \sum_{s \in S: f(s) = t} p_X(s)\) for discrete random variables or, \(p_Y(t) = \int_{s \in S: f(s) = t} p_X(s) ds\) for continuous random variables. Similarly, the PMF (or PDF) of \(Y^{\prime}\) can be expressed as: \(p_{Y^{\prime}}(t) = \sum_{s \in S: f(s) = t} p_{X^{\prime}}(s)\) for discrete random variables or, \(p_{Y^{\prime}}(t) = \int_{s \in S: f(s) = t} p_{X^{\prime}}(s) ds\) for continuous random variables. Since the PMFs (or PDFs) of \(X\) and \(X^{\prime}\) are essentially the same, we can rewrite the equations: \(p_Y(t) = \sum_{s \in S: f(s) = t} p_{X^{\prime}}(s)\) for discrete random variables, \(p_Y(t) = \int_{s \in S: f(s) = t} p_{X^{\prime}}(s) ds\) for continuous random variables. Now, comparing the PMFs (or PDFs) of \(Y\) and \(Y^{\prime}\), we see that they are equal: \(p_Y(t) = p_{Y^{\prime}}(t)\).

Step 3: Conclude that they have essentially the same distribution

Since the PMFs (or PDFs) of \(Y\) and \(Y^{\prime}\) are equal, we can conclude that \(f(X)\) and \(f\left(X^{\prime}\right)\) have essentially the same distribution.

Key Concepts

Probability Mass Function

The Probability Mass Function (PMF) is a foundational concept in understanding random variables, especially discrete ones. A PMF, denoted generally as \(p_X(x)\), quantifies the probability that a discrete random variable \(X\) is exactly equal to some value \(x\). For instance, when you roll a fair six-sided die, the probability of any specific side (like getting a 4) landing face up is \(1/6\). Here, the PMF for rolling a 4 is \(p_X(4)=1/6\).

The PMF is fundamental because it characterizes the distribution of discrete random variables completely. It has a few essential properties to remember:

• The probability for a specific outcome is always non-negative.
• The sum of probabilities of all possible outcomes is 1, which satisfies the principle of total probability.

When working with random variable transformation, understanding the PMF allows us to predict the behavior of a transformed variable, given the behavior of the original variable.

Probability Density Function

While the Probability Mass Function is used for discrete random variables, the Probability Density Function (PDF) applies to continuous random variables. The PDF, represented as \(p_X(x)\), describes the likelihood of a random variable \(X\) falling within a particular range of values. Unlike the PMF, where we find probabilities of exact values, the PDF gives us probabilities over intervals.

For example, under a normal (Gaussian) distribution, the probability of \(X\) being between \(a\) and \(b\) is obtained by the integral \(\int_{a}^{b} p_X(x) dx\). Here, the area under the PDF curve between \(a\) and \(b\) represents the probability of observing a value within that range. Key PDF properties include:

• The PDF itself is not a probability; it can be above 1 for certain ranges of \(X\).
• The area under the entire PDF curve across all possible values of \(X\) is exactly 1.

Understanding the PDF is crucial when dealing with continuous random variable transformations since it guides us in determining the new variable's distribution.

Random Variable Transformation

Transforming a random variable is a powerful tool in probability and statistics, allowing for the manipulation of variables to make them more analyzable or to fit a particular model. A transformation involves applying a function \(f\) to a random variable \(X\), resulting in a new random variable, say \(Y = f(X)\).

The transformed variable \(Y\) will have its own PMF or PDF, depending on whether it's discrete or continuous. The trick is to figure out what that distribution is based on the original variable's distribution and the transformation function applied. For discrete variables, this involves summing probabilities over the pre-images of values under \(f\), and for continuous variables, it involves a change-of-variable in integration. Through this process, we can explore a wide range of phenomena and make otherwise complex problems tractable.

Distribution Equality

Understanding distribution equality is central to making comparisons between different random variables or assessing the effects of transformations. When two random variables, \(X\) and \(X'\), are said to have essentially the same distribution, their PMFs or PDFs -- depending on whether they are discrete or continuous -- are identical. That is, for every point in their shared domain, the probability or density assigned by their distributions is the same.

As shown in the given exercise, if we apply the same transformation function \(f\) to both \(X\) and \(X'\), their resultant random variables \(Y = f(X)\) and \(Y' = f(X')\) also have the same distribution. This principle is critical when we're trying to understand the behavior of a random variable post-transformation, or when determining whether two scenarios are probabilistically equivalent.