Problem 35 - Applied Statistics and Probability for Engineers

Question

Suppose that \(f(x)=0.25\) for \(0

Step-by-Step Solution

Answer

The mean is 2, and the variance is \(\frac{4}{3}\).

Step 1: Understand the Distribution

The function given, \(f(x) = 0.25\) for \(0 < x < 4\), represents a uniform distribution over the interval \((0, 4)\). In a uniform distribution \(a\) and \(b\) are the boundaries of the interval, hence here \(a = 0\) and \(b = 4\).

Step 2: Calculate the Mean of X

For a uniform distribution over the interval \((a, b)\), the mean \(\mu\) is given by the formula: \[ \mu = \frac{a + b}{2} \]Substitute \(a = 0\) and \(b = 4\) to find:\[ \mu = \frac{0 + 4}{2} = 2 \]

Step 3: Calculate the Variance of X

The variance \(\sigma^2\) for a uniform distribution is given by:\[ \sigma^2 = \frac{(b - a)^2}{12} \]Substitute \(a = 0\) and \(b = 4\) to find:\[ \sigma^2 = \frac{(4 - 0)^2}{12} = \frac{16}{12} = \frac{4}{3} \]

Key Concepts

Understanding the Mean of Uniform Distribution

The mean of a uniform distribution is a simple concept. When you have a uniform distribution, all outcomes in the interval are equally likely. In a mathematical sense, this is evenly spread across the interval.
For a distribution defined over an interval aFor example, if your interval is from 0 to 4, you calculate it like this: Given by \(\mu = \frac{a + b}{2}\) a = 0 and b = 4,the mean, or average, is calculated as:
\[ \mu = \frac{0 + 4}{2} = 2 \].
This tells you that the center of the distribution, or the expected average outcome, is at the midpoint between your boundary values of 0 and 4.

Examining the Variance of Uniform Distribution

The variance in a uniform distribution tells you how spread out the values are. Variance provides a measure of how much the values deviate from the mean.
In the case of the uniform distribution, the formula to calculate variance is:\[ \sigma^2 = \frac{(b - a)^2}{12} \], where 'a' and 'b' are the boundaries.
The formula for the variance of a uniform distribution might seem complex, but it's primarily about checking how far apart the endpoints of the interval are, compared to 12.
For our interval from 0 to 4, the variance is calculated as:
\[ \sigma^2 = \frac{(4 - 0)^2}{12} = \frac{16}{12} = \frac{4}{3} \].
This tells you that the values of the uniform distribution vary around the mean by an amount given by the calculated variance.

The Basics of Probability Distribution

A probability distribution is essentially a map of all the possible outcomes of a random variable
and the likelihood of each outcome happening. In a uniform distribution, this map is particularly simple.
The outcomes are evenly spread across the interval, meaning each outcome is equally likely.
In mathematical terms, for a uniform distribution between given boundaries aFor the function \(f(x) = 0.25\) in the interval \(0 < x < 4\), each value is equally probable, creating a flat, rectangle-like distribution when graphed. This is why it's termed 'uniform'.
A uniform probability distribution helps you understand how the total probability or certainty of 1 is evenly allocated across the interval. This simplicity can be powerful, allowing for straightforward computations of statistics like mean and variance.