Problem 69 - Calculus and its applications

Question

For each probability density function, over the given interval, find \(\mathrm{E}(x), E\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{b-a}, \quad \text { over }[a, b] $$

Step-by-Step Solution

Answer

For the given uniform distribution, \(\text{E}(x) = \frac{a+b}{2}\), \(E(x^2) = \frac{a^2 + ab + b^2}{3}\), variance is \(\frac{(b-a)^2}{12}\), and standard deviation is \(\frac{b-a}{\sqrt{12}}\).

Step 1: Identify the Properties of the Uniform Distribution

The given function, \(f(x) = \frac{1}{b-a}\) over the interval \([a, b]\), represents a continuous uniform distribution. For a uniform distribution on \([a, b]\), it has a constant probability density function. The mean or expected value is given by \(\text{E}(x) = \frac{a+b}{2}\).

Step 2: Calculate E(x)

For a uniform distribution over the interval \([a, b]\), the expected value \(\text{E}(x)\) is calculated as:\[\text{E}(x) = \int_a^b x \frac{1}{b-a} \, dx = \frac{a+b}{2}\]

Step 3: Calculate E(x²)

To find \(E(x^2)\), calculate:\[\text{E}(x^2) = \int_a^b x^2 \frac{1}{b-a} \, dx = \frac{b^3 - a^3}{3(b-a)}\]Using the formula for uniform distribution, this simplifies to:\[\text{E}(x^2) = \frac{a^2 + ab + b^2}{3}\]

Step 4: Calculate the Variance

The variance is calculated as:\[\text{Var}(x) = E(x^2) - (E(x))^2\]Substitute \(E(x)\) and \(E(x^2)\) obtained from the previous steps:\[\text{Var}(x) = \frac{a^2 + ab + b^2}{3} - \left(\frac{a+b}{2}\right)^2\]\[\text{Var}(x) = \frac{(b-a)^2}{12}\]

Step 5: Calculate the Standard Deviation

The standard deviation is the square root of the variance. So:\[\text{SD}(x) = \sqrt{\text{Var}(x)} = \sqrt{\frac{(b-a)^2}{12}} = \frac{b-a}{\sqrt{12}}\]

Key Concepts

Probability Density Function

In the context of a uniform distribution, the probability density function (PDF) plays a crucial role in determining the likelihood of outcomes within a specific range. For a uniform distribution on the interval `[a, b]`, the PDF is expressed as: \[ f(x) = \frac{1}{b-a} \] This simple formula reflects that each possible outcome within the range [a, b] is equally likely. Thus, the PDF is a useful tool for modeling situations where there's no preference or bias towards any particular outcome within the specified range. For example, the roll of a fair die is a classic real-world approximation of a uniform distribution. Moreover, since the PDF integrates to 1 over the interval `[a, b]`, it confirms the fundamental principle of probability that the total probability of all possible outcomes equals 1. This uniformity makes calculations straightforward, especially when determining probabilities for continuous random variables.

Expected Value

The expected value, often referred to as the mean, represents the center or average of a probability distribution. For a uniform distribution over the interval `[a, b]`, the expected value \( \mathrm{E}(x) \) is computed by: \[ \mathrm{E}(x) = \frac{a+b}{2} \] This is derived by taking an integral of the product of the variable and the PDF, which comes out as the midpoint of the interval `[a, b]`. This calculation shows that for a uniform distribution, the mean is simply the average of the two endpoints. It's worth noting that in real-world applications, the expected value assists in predicting future outcomes based on the distribution of past data. By knowing the expected value, one can identify the most probable value within the distribution, serving as a reliable indicator of central tendency.

Variance

Variance measures the extent of variability or spread in a set of values, indicating how much individual data points deviate from the mean. For a uniform distribution over the interval `[a, b]`, the variance \( \text{Var}(x) \) is given by: \[ \text{Var}(x) = \frac{(b-a)^2}{12} \] To calculate variance, one takes the expected value of the square \( E(x^2) \) and subtracts the square of the expected value \( (E(x))^2 \). Variance thus encapsulates the distribution's spread, with a larger variance indicating greater variance from the mean. In practical terms, understanding variance helps in assessing the reliability of predictions and the risk associated with potential outcomes. Lower variance implies that the data points are closely concentrated around the mean, suggesting high consistency.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values and is directly related to variance. For a uniform distribution over the interval `[a, b]`, the standard deviation \( \text{SD}(x) \) is expressed as: \[ \text{SD}(x) = \sqrt{\text{Var}(x)} = \frac{b-a}{\sqrt{12}} \] The standard deviation is simply the square root of the variance, translating the variance units back into the original data units, making it a bit more intuitive. It provides a clear understanding of how much individual observations deviate from the expected value. Real-world significance of standard deviation includes its use in finance to gauge the volatility of an asset, in operations to track process consistency, and in quality control to monitor product reliability. A smaller standard deviation signifies less variability and implies that the data points are close to the mean, fostering predictability.