Uniform Distribution
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This post covers Introduction to probability from Statistics for Engineers and Scientists by William Navidi.
Basic Ideas
The Uniform Distribution
The probability density function of the continuous uniform distribution with
parameters $a$ and $b$ is \(p(x) = P(X = x) = \left\{ \begin{array}{ll} \frac{1}{b-a}& \mbox{if}~ a \lt x \lt b \\ 0 & otherwise \end{array} \right.\)
If $X$ is a random variable with probability density function $f (x)$, we say that $X$ is
uniformly distributed on the interval $(a, b)$.
Let $X \sim U(a, b)$. Then
$ 𝜇_{X} = \frac{1}{ 2 (a + b) } $
$ 𝜎^{2}_{X}= \frac{(b − a)^{2}}{12} $
Because the probability density function for a uniform random variable is constant over the range of possible values, probabilities for uniform random variables generally involve areas of rectangles, which can be computed without integrating
When a motorist stops at a red light at a certain intersection, the waiting time for the light to turn green, in seconds, is uniformly distributed on the interval $(0, 30)$. Find the probability that the waiting time is between $10$ and $15$ seconds.