Problem 9 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(U\) be a random variable with a \(U(\alpha, \beta)\) distribution. a. Determine the expectation of \(U\). b. Determine the variance of \(U\).

Step-by-Step Solution

Answer

Expectation: \(\frac{\alpha + \beta}{2}\), Variance: \(\frac{(\beta - \alpha)^2}{12}\).

Step 1: Understanding Uniform Distribution

The random variable \(U\) follows a continuous uniform distribution between \(\alpha\) and \(\beta\), denoted as \(U(\alpha, \beta)\). This means that \(U\) is equally likely to take any value in the interval \([\alpha, \beta]\).

Step 2: Expectation of Uniform Distribution

For a continuous uniform distribution \(U(\alpha, \beta)\), the expectation is given by the formula:\[ E(U) = \frac{\alpha + \beta}{2} \]This formula calculates the midpoint of the interval, which is the expected value.

Step 3: Calculate the Expectation

Using the expectation formula:\[ E(U) = \frac{\alpha + \beta}{2} \]Substitute the values \(\alpha\) and \(\beta\) (as given in specific scenarios) to find the expectation.

Step 4: Variance of Uniform Distribution

The variance of a continuous uniform distribution \(U(\alpha, \beta)\) is calculated using the formula:\[ \text{Var}(U) = \frac{(\beta - \alpha)^2}{12} \]This formula measures the dispersion of the distribution.

Step 5: Calculate the Variance

Substitute \(\alpha\) and \(\beta\) into the variance formula:\[ \text{Var}(U) = \frac{(\beta - \alpha)^2}{12} \]Calculate the result to find the variance.

Key Concepts

Expectation

In probability theory, the expectation, or expected value, of a random variable is a key concept that provides the average or mean value of the variable over many trials. For a continuous uniform distribution like our variable \(U\), which is uniformly distributed over an interval \([\alpha, \beta]\), the expectation is calculated using the simple formula

• \(E(U) = \frac{\alpha + \beta}{2}\).

This formula gives us the midpoint of the interval, essentially telling us that if you randomly pick a point in this interval, on average, the point will be the midpoint. The expectation doesn't predict the result of a single trial, but it's extremely useful for understanding what to generally expect over many trials.

Variance

Variance is another fundamental concept in statistics that helps us understand how much the values of a random variable, such as \(U\), deviate from the expectation on average. In the context of a continuous uniform distribution, the variance is calculated with:

• \(\text{Var}(U) = \frac{(\beta - \alpha)^2}{12}\).

This formula quantifies the spread or dispersion of the distribution. A larger interval \([\alpha, \beta]\) means a higher variance since there is a greater spread of potential values. Knowing what the variance is can help us understand the degree of uncertainty associated with the random variable—not just the average, but how much it varies around that average.

Random Variable

A random variable is an essential concept in probability and statistics. It is a variable that takes on different values based on the outcome of a random phenomenon. In this exercise, \(U\) is our random variable, representing a range of outcomes within the interval \([\alpha, \beta]\).

• Random variables can be discrete (taking on a countable number of values) or continuous (taking on any value within a range).
• \(U\) is continuous since it can assume any real value between \(\alpha\) and \(\beta\).

Understanding the nature of a random variable helps us model and analyze various real-world scenarios, from simple dice rolls to complex scenarios like stock prices. Random variables are a foundational element in both probability theory and statistical modeling.

Continuous Distribution

Continuous distribution describes a random variable that can take an infinite number of values within a given range. The key feature is that these values are not countable and can encompass any number in a specific interval.

• An example of a continuous distribution is the uniform distribution \(U(\alpha, \beta)\).
• This distribution implies that every value in the interval \([\alpha, \beta]\) has an equal chance of being observed.

Continuous distributions are integral in various real-life applications, such as measuring the height of individuals or analyzing time-based processes. Understanding continuous distributions allows us to apply calculus-based techniques to calculate probabilities and make predictions, significantly enhancing our analytical abilities in handling real-world randomness.