Problem 16 - A Computational Introduction to Number Theory and Algebra

Question

Let \(X\) and \(Y\) be independent random variables, where \(X\) is uniformly distributed over a set \(S,\) and \(Y\) is uniformly distributed over a set \(T \subseteq S\). Define a third random variable \(Z\) as follows: if \(X \in T,\) then \(Z:=X\); otherwise, \(Z:=Y\). Show that \(Z\) is uniformly distributed over \(T\).

Step-by-Step Solution

Answer

Question: Prove that if \(X\) and \(Y\) are independent random variables where \(X\) is uniformly distributed over set \(S\) and \(Y\) is uniformly distributed over set \(T \subseteq S\), then the random variable \(Z\), defined as \(Z := X\) if \(X \in T\) and \(Z := Y\) otherwise, is uniformly distributed over \(T\). Answer: To prove that \(Z\) is uniformly distributed over \(T\), we computed the probability density function (pdf) of \(Z\) and showed that it is equal to the pdf of \(Y\) for all \(z \in T\), which is given by \(f_Z(z) = \frac{1}{|T|}\). This means that \(Z\) has the same pdf as of \(Y\) and is therefore uniformly distributed over \(T\).

Step 1: Description of the random variables

Let's first describe the given random variables: - \(X\) is uniformly distributed over set \(S\). This means that for any \(x \in S\), the probability density function (pdf) of \(X\) is given by \(f_X(x) = \frac{1}{|S|}\), where \(|S|\) is the number of elements in set \(S\). - \(Y\) is uniformly distributed over set \(T\). This means that for any \(y \in T\), the probability density function (pdf) of \(Y\) is given by \(f_Y(y) = \frac{1}{|T|}\), where \(|T|\) is the number of elements in set \(T\). - \(Z\) is defined as \(Z := X\) if \(X \in T\) and \(Z := Y\) otherwise. Our goal is to show that \(Z\) is uniformly distributed over \(T\).

Step 2: Compute the probability density function of \(Z\)

To compute the probability density function of \(Z\), we need to consider two cases: when \(Z := X\) and \(Z := Y\). Case 1: \(Z = X\) The probability that \(Z = X\) is equal to the probability that \(X \in T\). Since \(X\) is uniformly distributed over \(S\), the probability that \(X \in T\) is given by: \(P(X \in T) = \frac{|T|}{|S|}\). Case 2: \(Z = Y\) The probability that \(Z = Y\) is equal to the probability that \(X \notin T\). This probability is given by: \(P(X \notin T) = 1 - P(X \in T) = 1 - \frac{|T|}{|S|}\). Now, we can calculate the pdf of \(Z\).

Step 3: Determine the pdf of \(Z\)

To sum up, Z is uniformly distributed over set \(T\) if the pdf of \(Z\) is equal to the pdf of \(Y\). To prove that \(Z\) is uniformly distributed over \(T\), we can find the pdf of \(Z\) given the probabilities that we computed in step 2. For any \(z \in T\), we have: $$ f_Z(z) = P(Z = z)\\ = P(Z = X | X \in T)P(X \in T) + P(Z = Y | X \notin T)P(X \notin T) $$ Using the probabilities we computed in step 2 and the pdf's of \(X\) and \(Y\), we find: $$ f_Z(z) = \frac{1}{|S|}\frac{|T|}{|S|} + \frac{1}{|T|}\left(1 - \frac{|T|}{|S|}\right)\\ = \frac{1}{|T|} $$ Since this is true for all \(z \in T\), this means that \(Z\) has the same pdf as of \(Y\) and thus is uniformly distributed over \(T\).

Key Concepts

Uniform Distribution

The concept of a Uniform Distribution is quite simple yet crucial in the study of probability and statistics. When we say that a random variable, like variable \(X\) in our exercise, follows a Uniform Distribution over a set \(S\), it means that each possible outcome within that set has an equal probability of occurring. Imagine rolling a fair die. Each side, numbered one through six, has the same chance of landing face up, which illustrates the fundamental nature of Uniform Distribution.

Mathematically, if \(X\) is uniformly distributed over set \(S\), the probability density function is

• \(f_X(x) = \frac{1}{|S|}\)

Here, \(|S|\) denotes the total number of elements in set \(S\), and each element \(x\) in \(S\) is equally likely to be chosen. Uniform Distribution is powerful because it simplifies complex situations by making predictions straightforward. This is precisely why, in our exercise, \(Y\) is also said to be uniformly distributed over a subset \(T\), following the same logic. Conceptually, Uniform Distribution offers a baseline for understanding randomness in defined spaces.

Probability Density Function

The Probability Density Function (pdf) forms the backbone of continuous probability distributions, providing a function that describes the likelihood of a random variable taking on various possible values. In a Uniform Distribution, since every outcome is equally likely, the pdf is a constant function over the distribution range.

To understand pdf better, consider a simple example of a spinner divided equally into sections numbered 1 to 10. The pdf of the spinner's result is

• \(f(x) = \frac{1}{10}\)

because there are 10 equally likely outcomes. The area under the pdf across its range is always equal to 1, indicating

• Total probability is 1.

In the case of our random variables \(X\) and \(Y\), the pdf helps us calculate various probabilities and make clear predictions. Especially when dealing with independent random variables, knowing the pdf allows us to treat each variable separately before combining them, as seen in step-by-step solutions when determining the attributes of a derived variable like \(Z\). The pdf is an essential tool for quantifying and working with probabilities.

Independent Random Variables

Independent Random Variables are a core concept in probability theory. When two or more random variables are independent, it means that the occurrence or outcome of one does not influence the other. In the context of our exercise, \(X\) and \(Y\) are independent random variables. This independence is critical as it simplifies many calculations and implies that actions affecting one variable do not affect the other.

Consider flipping two separate coins. Whether the first coin lands heads or tails doesn't impact the outcome of the second coin. This lack of influence between the variables allows us to express combined probabilities in a straightforward way, such as multiplying the probabilities of individual events. This principle is useful, for instance, when defining or analyzing a new random variable \(Z\) as a combination of \(X\) and \(Y\).

• For independent events, if \(A\) and \(B\) are two events, the probability of both occurring is \(P(A \cap B) = P(A) \times P(B)\).

The independence assumption significantly simplifies our work with random variables, letting us analyze them separately, leverage their individual distributions, and make predictions about new variables derived from them, as shown in our detailed step-by-step solution with \(Z\). Understanding independence is a keystone of analyzing randomness and probability in complex systems.