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.