Problem 2 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Suppose \(X\) has a uniform distribution over the points \(\\{1,2,3,4,5,6\\}\) and that \(g(x)=\sin \left(\frac{\pi}{2} x\right)\). a. Determine the distribution of \(Y=g(X)=\sin \left(\frac{\pi}{2} X\right)\), that is, specify the values \(Y\) can take and give the corresponding probabilities. b. Let \(Z=\cos \left(\frac{\pi}{2} X\right)\). Determine the distribution of \(Z\). c. Determine the distribution of \(W=Y^{2}+Z^{2}\). Warning: in this example there is a very special dependency between \(Y\) and \(Z\), and in general it is much harder to determine the distribution of a random variable that is a function of two other random variables. This is the subject of Chapter 11 .

Step-by-Step Solution

Answer

a. \(Y\) has values \{-1,0,1\} with probabilities \(\frac{1}{6}, \frac{1}{2},\frac{1}{3}\). b. \(Z\) has values \{-1,0,1\} with probabilities \(\frac{1}{3}, \frac{1}{2}, \frac{1}{6}\). c. \(W=1\) with probability 1.

Step 1: Understand the Uniform Distribution of X

Since \(X\) is uniformly distributed over the set \(\{1,2,3,4,5,6\}\), it means that \(X\) can take any of these six values with equal probability. Each value therefore has a probability of \(\frac{1}{6}\).

Step 2: Calculate g(X) = \(\sin\left(\frac{\pi}{2} X\right)\)

Calculate \(g(x) = \sin\left(\frac{\pi}{2} x\right)\) for each \(x\) in \(\{1, 2, 3, 4, 5, 6\}\):- \(x = 1\): \(g(1) = \sin\left(\frac{\pi}{2}\right) = 1\)- \(x = 2\): \(g(2) = \sin\left(\pi\right) = 0\)- \(x = 3\): \(g(3) = \sin\left(\frac{3\pi}{2}\right) = -1\)- \(x = 4\): \(g(4) = \sin\left(2\pi\right) = 0\)- \(x = 5\): \(g(5) = \sin\left(\frac{5\pi}{2}\right) = 1\)- \(x = 6\): \(g(6) = \sin\left(3\pi\right) = 0\)

Step 3: Determine Distribution of Y

From the calculations, \(Y = g(X)\) can take the values \(\{-1, 0, 1\}\). We find the probabilities by counting the occurrences of each \(Y\) value:- \(Y = 1\) occurs when \(X = 1\) or \(X = 5\), so probability is \(\frac{2}{6} = \frac{1}{3}\).- \(Y = 0\) occurs when \(X = 2\), \(X = 4\), or \(X = 6\), so probability is \(\frac{3}{6} = \frac{1}{2}\).- \(Y = -1\) occurs when \(X = 3\), so probability is \(\frac{1}{6}\).

Step 4: Calculate h(X) = \(\cos\left(\frac{\pi}{2} X\right)\)

Calculate \(h(x) = \cos\left(\frac{\pi}{2} x\right)\) for each \(x\) in \(\{1, 2, 3, 4, 5, 6\}\):- \(x = 1\): \(h(1) = \cos\left(\frac{\pi}{2}\right) = 0\)- \(x = 2\): \(h(2) = \cos\left(\pi\right) = -1\)- \(x = 3\): \(h(3) = \cos\left(\frac{3\pi}{2}\right) = 0\)- \(x = 4\): \(h(4) = \cos\left(2\pi\right) = 1\)- \(x = 5\): \(h(5) = \cos\left(\frac{5\pi}{2}\right) = 0\)- \(x = 6\): \(h(6) = \cos\left(3\pi\right) = -1\)

Step 5: Determine Distribution of Z

From the calculations, \(Z = h(X)\) can take the values \(\{-1, 0, 1\}\). We find the probabilities by counting the occurrences of each \(Z\) value:- \(Z = 0\) occurs when \(X = 1\), \(X = 3\), or \(X = 5\), so probability is \(\frac{3}{6} = \frac{1}{2}\).- \(Z = -1\) occurs when \(X = 2\) or \(X = 6\), so probability is \(\frac{2}{6} = \frac{1}{3}\).- \(Z = 1\) occurs when \(X = 4\), so probability is \(\frac{1}{6}\).

Step 6: Evaluate W = Y^2 + Z^2 for Each X

Calculate \(W = Y^2 + Z^2\) for each value of \(X\):- \(X = 1\): \(W = 1^2 + 0^2 = 1\)- \(X = 2\): \(W = 0^2 + (-1)^2 = 1\)- \(X = 3\): \(W = (-1)^2 + 0^2 = 1\)- \(X = 4\): \(W = 0^2 + 1^2 = 1\)- \(X = 5\): \(W = 1^2 + 0^2 = 1\)- \(X = 6\): \(W = 0^2 + (-1)^2 = 1\)

Step 7: Distribution of W

Since for each value of \(X\), \(W\) is calculated to be 1, it means \(W\) can take only one value, which is 1. Therefore, \(P(W=1) = 1\).

Key Concepts

Probability Distribution

A probability distribution describes how the values of a random variable are distributed. It provides a full picture of how probabilities are spread over possible outcomes. In the case of a uniform distribution, each outcome is equally likely.
For example, if we have a set \(\{1, 2, 3, 4, 5, 6\}\), and each number is equally likely to be chosen, then the probabilities are the same for each number. Specifically, the probability of any particular number is \(\frac{1}{6}\), since there are six outcomes.
This concept is crucial because it helps to understand how variables behave and assists in predicting outcomes under uncertainty. The uniform distribution is a particular case that simplifies calculations, often used in introductory statistics and probability.

Random Variables

Random variables serve as tools to connect outcomes from a random process with numerical quantities. In essence, a random variable assigns a number to each outcome in a sample space. There are two types: discrete and continuous.
For instance, in the example exercise, \(X\) is a discrete random variable because it can only take the specific values \(\{1, 2, 3, 4, 5, 6\}\). Each value relates to an equal chance in the uniform distribution context.
Function transformations, like \(g(X)\) or \(h(X)\), use functions to map the values of random variables into other numerical outcomes, introducing new distributions such as for \(Y\) and \(Z\). Understanding random variables is key to exploring advanced topics, such as joint distributions and dependency between variables.

Trigonometric Functions

Trigonometric functions relate angles to ratios of triangle sides and play significant roles in various fields, such as physics and engineering. In probability, they help transform random variables.
For example, in our exercise, we use \( \sin(\frac{\pi}{2}X) \) and \( \cos(\frac{\pi}{2}X) \) to create new random variables \(Y\) and \(Z\). These transform the uniform distribution of \(X\) into distributions based on the sine and cosine values.
These functions cycle through values as inputs (like \(X\)'s values) evolve, introducing periodic behavior. This cyclic property can simplify calculations under certain conditions.

• \( \sin \) and \( \cos \) oscillate between -1 and 1.
• They have a period of \(2\pi\), which affects transformations.

Thus, using trigonometric functions with random variables is effective for analyzing oscillatory or periodic phenomena.