SciTech-Mathmatics-Probability+Statistics: Distribution :

The Uniform Distribution

BY ZACH BOBBITTPOSTED ON MARCH 2, 2021
The uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to occur.

If a random variable \(\large X\) follows a uniform distribution,
then the probability that \(\large X\) takes on a value between \(\large x_1\) and \(\large x_2\) can be found by the following formula:

\(\large \begin{array}{lrl} \\ &P(x_1 < X < x_2) =& \frac{x_2\ –\ x_1}{b\ –\ a} \\ where:& & \\ & x_1: & the\ \bm{lower\ value}\ of\ \bm{interest} \\ & x_2: & the\ \bm{upper\ value}\ of\ \bm{interest} \\ & a: & the\ \bm{minimum\ possible\ value} \\ & b: & the\ \bm{maxmum\ possible\ value} \\ \end{array}\)

For example, suppose the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds.

If we randomly select a dolphin at random, we can use the formula above to determine the probability that the chosen dolphin will weigh between 120 and 130 pounds:

P(120 < X < 130) = (130 – 120) / (150 – 100)
P(120 < X < 130) = 10 / 50
P(120 < X < 130) = 0.2
The probability that the chosen dolphin will weigh between 120 and 130 pounds is 0.2.