91Ó°ÊÓ

In a continuous uniform distribution, every outcome in a given interval is equally likely to occur. Let's focus on calculating the mean of a uniform distribution, which is a vital measure of central tendency. For a uniform distribution defined over the interval \([a, b]\), the mean or expected value \(\mu\) can be found using the straightforward formula:\[\mu = \frac{a+b}{2}\]This formula tells us that the mean is simply the midpoint of the interval.
It's easy to remember because you're just averaging the lower and upper bounds.

• In the exercise, we have \(a = -1\) and \(b = 1\).
• Thus, \(\mu = \frac{-1+1}{2} = 0\).

This is because the symmetry of the interval around zero means the average will logically be zero. Understanding the mean helps provide insight into the balance point of the distribution.

The variance of a uniform distribution measures how much values deviate from the mean.
It's a key indicator of the distribution's spread. For a uniform distribution on an interval \([a, b]\), variance \(\sigma^2\) is calculated with the formula:\[\sigma^2 = \frac{(b-a)^2}{12}\]Here's how to interpret it:

• The term \(b-a\) is the interval's length, indicating the amount of total spread.
• Dividing by 12 gives a measure of average squared deviation from the mean.

Applying this to our interval of \([-1, 1]\), the calculation proceeds as:

• Length of the interval \((b-a) = 2\).
• Variance \(\sigma^2 = \frac{2^2}{12} = \frac{4}{12} = \frac{1}{3}\).

Understanding variance helps assess how spread out the distribution is from the mean, complementing the information given by the mean itself.

The cumulative distribution function (CDF) of a uniform distribution shows how probabilities accumulate over the interval.
It represents the probability that a random variable \(X\) will take a value less than or equal to \(x\). For our uniform distribution over the interval \([-1, 1]\), the CDF \(F(x)\) is defined piecewise as follows:\[F(x) = \begin{cases} 0, & x < -1, \\frac{x+1}{2}, & -1 \leq x \leq 1, \1, & x > 1.\end{cases}\]This piecewise function explains:

• For \(x < -1\), the probability is 0 since \(x\) falls outside the interval.
• For \(-1 \leq x \leq 1\), the probabilities accumulate linearly. The transformation \(\frac{x+1}{2}\) accounts for the proportion of the way through the interval \([-1, 1]\).
• For \(x > 1\), the probability is 1, covering all possible outcomes.

The CDF is invaluable for determining probabilities over ranges of values within a distribution, giving you a full picture of the distribution dynamics.