Problem 13 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(W\) have a \(U(\pi, 2 \pi)\) distribution. What is larger: \(\mathrm{E}[\sin (W)]\) or \(\sin (\mathrm{E}[W])\) ? Check your answer by computing these two numbers.

Step-by-Step Solution

Answer

\(\mathrm{E}[\sin(W)]\) is larger than \(\sin(\mathrm{E}[W])\).

Step 1: Understanding the Distribution

We are given that the random variable \(W\) follows a uniform distribution on the interval \([\pi, 2\pi]\). This means that any value within this interval is equally likely to occur.

Step 2: Compute the Expected Value of W

For a uniform distribution \(U(a, b)\), the expected value \(\mathrm{E}[W]\) is calculated as the midpoint of the interval. Thus, \(\mathrm{E}[W] = \frac{\pi + 2\pi}{2} = \frac{3\pi}{2}\).

Step 3: Compute \(\sin(\mathrm{E}[W])\)

Substitute \(\mathrm{E}[W] = \frac{3\pi}{2}\) into the sine function: \(\sin(\mathrm{E}[W]) = \sin\left(\frac{3\pi}{2}\right) = -1\).

Step 4: Compute \(\mathrm{E}[\sin(W)]\)

The expectation \(\mathrm{E}[\sin(W)]\) for a uniform distribution can be calculated by integrating the sine function over the interval \([\pi, 2\pi]\). We find:\[\mathrm{E}[\sin(W)] = \frac{1}{2\pi - \pi} \int_{\pi}^{2\pi} \sin(x) \, dx\]Evaluate the integral:\[ = \frac{1}{\pi} \left[-\cos(x) \right]_\pi^{2\pi}\]\[ = \frac{1}{\pi} \left[ -\cos(2\pi) + \cos(\pi) \right]\]\[ = \frac{1}{\pi} \left[ -(1) + (-1) \right] = -\frac{2}{\pi}\]

Step 5: Compare the Two Values

We found that \(\mathrm{E}[\sin(W)] = -\frac{2}{\pi}\) and \(\sin(\mathrm{E}[W]) = -1\). Since \(-\frac{2}{\pi} \approx -0.6366\), which is greater than \(-1\), we conclude that \(\mathrm{E}[\sin(W)]\) is larger than \(\sin(\mathrm{E}[W])\).

Key Concepts

Expected Value

When dealing with probability distributions, the Expected Value is a key concept. It represents the average or mean of a random variable over a vast number of trials. For a uniform distribution, this is calculated as the midpoint of the interval. This is because each outcome within the interval is equally likely. So, if a random variable, say \(W\), ranges from \(\pi\) to \(2\pi\), its Expected Value would be:

• The midpoint: \(\frac{\pi + 2\pi}{2} = \frac{3\pi}{2}\)

In this scenario, the Expected Value tells us the central tendency of \(W\) when observed over many occurrences.

The Expected Value is useful because it simplifies comparisons. For example, when analyzing transformations of variables, like applying a sine function to our uniform distribution, knowing the expected value beforehand makes the computation easier.

Sine Function

The Sine Function is a trigonometric function that can be represented as \(\sin(x)\), where \(x\) is an angle measured in radians. The sine function has an output range from -1 to 1, making it periodic with a cycle repeating every \(2\pi\). Knowing how it behaves is crucial when dealing with angles that return values outside the basic sine cycle.

For example, \(\sin\left(\frac{3\pi}{2}\right) = -1\). This can be visualized on the sine curve where \(\frac{3\pi}{2}\) corresponds to the lowest point on the curve, giving the minimum value of -1. The sinusoidal nature of this function often makes it challenging yet interesting to interact with in probability and calculus, especially when integrated.

Integration

Integration is a fundamental concept in calculus used for finding areas under curves, among other applications. When working with random variables, integration helps determine the expected values of more complex transformations.

• In the given problem, integrating the sine function from \(\pi\) to \(2\pi\) helps compute the expectation: \(\mathrm{E}[\sin(W)]\).

This integral is formulated as:

• \( \int_{\pi}^{2\pi} \sin(x) \, dx \)

Evaluating this integral leads to the average value of \(\sin(W)\) within the given interval, resulting in \(-\frac{2}{\pi}\). This result provides insight into how the sine function behaves across the interval from \(\pi\) to \(2\pi\) and assists in comparing it with other transformations of \(W\). Integration thus serves as a crucial tool in the determination of expected values, particularly when dealing with non-linear transformations like the sine function.