91Ó°ÊÓ

In a uniform distribution, calculating percentiles involves understanding the cumulative distribution function (CDF). Percentiles are a statistical measure indicating the value below which a given percentage of observations fall. To find the - **Percentile Expression**: The percentile of a uniform distribution on is the solution to the equation where CDF equals the desired percentile. The CDF, given by represents the probability that the variable takes a value less than or equal to For the find the value that satisfies:\[ \frac{x_p - A}{B - A} = p\]Solving for we get:\[ x_p = p(B - A) + A\]This formula allows you to find any percentile by plugging in the desired probability value. It highlights the linearity of the CDF in uniform distributions.

Expectation or the expected value is a fundamental concept in probability theory. For a uniform distribution, expectation gives us the "center" or average value that the random variable takes. This is most intuitively understood as the balance point of the probability distribution.- **Expectation Formula**: The expectation of a uniform distribution on is the midpoint of the interval. It is calculated as:\[ E(X) = \frac{A + B}{2}\]This formula tells us that the expected value is simply the mean of the lower and upper bounds of the distribution. It provides a simple but powerful measure of central tendency for any uniform distribution.

Variance and standard deviation measure the spread or variability of a distribution. For a uniform distribution, these parameters help us understand how data is dispersed from the mean.- **Variance**: The variance captures how data varies around the mean. It's calculated as:\[ V(X) = \frac{(B - A)^2}{12}\]This formula shows the proportionality to the square of the interval length. A wider distribution results in greater variance.- **Standard Deviation**: The standard deviation is the square root of the variance, providing a more intuitive measure of spread:\[ \sigma_{X} = \frac{B - A}{\sqrt{12}}\]Standard deviation is often preferred because it's in the same units as the original data, making it easier to interpret.

In probability, integration is a tool used to calculate expectations for more complex situations, like when dealing with powers of a random variable. This involves integrating the probability density function (PDF) over the specified interval.- **Expectation of Powers**: To find the expectation of a random variable to the power of (denoted as in a uniform distribution from we use the integral:\[ E(X^n) = \int_A^B x^n \cdot \frac{1}{B - A} \, dx\]Evaluating this integral gives:\[ E(X^n) = \frac{B^{n+1} - A^{n+1}}{(n+1)(B-A)}\]This formula allows us to compute the expected value of any integer power of . It highlights that as increases, the expectation becomes more influenced by the endpoints of the interval, underscoring the importance of integration in finding expectations of complex random variables.