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Chapter 4: Problem 7

Let \(X\) have a uniform distribution on the interval \([A, B]\). a. Obtain an expression for the \((100 p)\) th percentile. b. Compute \(E(X), V(X)\), and \(\sigma_{X}\). c. For \(n\), a positive integer, compute \(E\left(X^{n}\right)\).

Short Answer

Expert verified
Percentile: \(A + p(B-A)\), \(E(X) = \frac{A + B}{2}\), \(V(X) = \frac{(B-A)^2}{12}\), \(\sigma_X = \frac{B-A}{\sqrt{12}}\), \(E(X^n) = \frac{1}{n+1}\frac{B^{n+1}-A^{n+1}}{B-A}\).

Step by step solution

01

Understanding the Uniform Distribution

The uniform distribution is defined as having constant probability over an interval. For a continuous uniform distribution on the interval \([A, B]\), the probability density function (PDF) is given by \(f(x) = \frac{1}{B-A}\) for \(A \leq x \leq B\).
02

Finding the Percentile

To find the \((100p)\)th percentile of a uniform distribution, use the formula for the cumulative distribution function (CDF), which is \(F(x) = \frac{x-A}{B-A}\). Set \(F(x) = p\), solve for \(x\) to find the percentile \(P = A + p(B-A)\).
03

Calculating Expected Value

The expected value \(E(X)\) for a uniform distribution on the interval \([A, B]\) is calculated using the formula \(E(X) = \frac{A + B}{2}\).
04

Calculating Variance

The variance \(V(X)\) is calculated with the formula \(V(X) = \frac{(B-A)^2}{12}\) for a uniform distribution on the interval \([A, B]\).
05

Calculating Standard Deviation

The standard deviation \(\sigma_X\) is simply the square root of the variance. Therefore, \(\sigma_X = \sqrt{\frac{(B-A)^2}{12}} = \frac{B-A}{\sqrt{12}}\).
06

Computing General Expected Value for Powers of X

For \(E(X^n)\), calculate using the integral: \(E(X^n) = \int_A^B x^n f(x) \, dx\). Since \(f(x) = \frac{1}{B-A}\), \(E(X^n) = \frac{1}{B-A} \int_A^B x^n \, dx \). Evaluating this integral gives \(E(X^n) = \frac{1}{n+1} \left( B^{n+1} - A^{n+1} \right)\) divided by \(B-A\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Percentile
A percentile indicates the value below which a given percentage of observations in a group fall. In the context of a uniform distribution, calculating the \(100p\)th percentile involves using the cumulative distribution function (CDF). This function, denoted as \(F(x)\), helps determine the position or rank of a value within a specified range.
For a uniform distribution on the interval \([A, B]\), the CDF is given by \(F(x) = \frac{x-A}{B-A}\). To find the \(100p\)th percentile, you set the CDF equal to \(p\) and solve for \(x\). The result is the formula: \[x = A + p(B-A)\].
This formula shows how the percentile is calculated by adjusting the lower limit \(A\) by the product of the probability \(p\) and the range \(B - A\). Percentiles are widely used in statistics to understand distribution trends and anomalies.
Expected Value
The expected value, often represented as \(E(X)\), is essentially the mean or average of a random variable. For a uniform distribution, it's the value around which the data points cluster. Calculating the expected value provides a measure of center for the distribution.
For a uniform distribution on the interval \([A, B]\), the formula for expected value is \[E(X) = \frac{A+B}{2}\].
This formula indicates that the expected value is simply the midpoint of the interval. This makes sense because, in a uniform distribution, every value between \(A\) and \(B\) is equally likely, so the average is the center of the interval. Understanding the expected value gives insights into the overall behavior of the distribution.
Variance
Variance measures the spread or variability of a dataset. It tells us how much the values of a random variable differ from the mean. In other words, it quantifies the expected degree of variation.
For a uniform distribution over the interval \([A, B]\), the variance is determined using the formula \[V(X) = \frac{(B-A)^2}{12}\].
This formula highlights that variance is related to the square of the length of the interval, divided by 12. The longer the interval, the greater the spread of values around the expected value. Variance is a crucial concept because it impacts other statistical measures like standard deviation.
Standard Deviation
The standard deviation \(\sigma_X\) is a measure of the amount of variation or dispersion of a set of values. It is derived from the variance and gives context to the dispersion in the same units as the data, making it more interpretable.
The standard deviation for a uniform distribution can be found by taking the square root of the variance. Thus, the formula is: \[\sigma_X = \sqrt{\frac{(B-A)^2}{12}} = \frac{B-A}{\sqrt{12}}\].
This shows that the standard deviation is proportionate to the interval width \(B - A\), providing insight into how concentrated or spread out the data points are relative to the mean. Understanding standard deviation helps in comparing variability between different datasets and distributions.

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