6.9:
Uniform Distribution
The uniform distribution is a continuous probability distribution associated with events that are equally likely to occur.
Its probability density is expressed by a rectangular function where 'a' and 'b' are the lower and upper cut-offs, respectively.
For example, the voltage provided by the electricity company is uniformly distributed, say, between 122 and 126 volts.
In this case, the probability density is plotted as a function of the voltage supplied.
The total area under the graph should always be one. Since the range is 4 volts, the height must be one divided by 4.
One might wonder what is the probability of any household getting a voltage less than 123 volts?
It can be found from the area under the segment, which is the product of the width and height of the section.
This mean voltage supplied is the sum of the cut-off values divided by two, which in this case is a 124 volts.
The standard deviation is given by the range divided by the square root of twelve, which is found to be 1.2 volts.
6.9:
Uniform Distribution
The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.
Two essential properties of this distribution are
- The area under the rectangular shape equals 1.
- There is a correspondence between the probability of an event and the area under the curve.
Further, the mean and standard deviation of the uniform distribution can be calculated when the lower and upper cut-offs, denoted as a and b, respectively, are given. For a random variable x, in a uniform distribution, given a and b, the probability density function is f(x) is calculated as
Consider data of 55 smiling times, in seconds, of an eight-week-old baby:
10.4, 19.6, 18.8, 13.9, 17.8, 16.8, 21.6, 17.9, 12.5, 11.1, 4.9, 12.8, 14.8, 22.8, 20.0, 15.9, 16.3, 13.4, 17.1, 14.5, 19.0, 22.8, 1.3, 0.7, 8.9, 11.9, 10.9, 7.3, 5.9, 3.7, 17.9, 19.2, 9.8, 5.8, 6.9, 2.6, 5.8, 21.7, 11.8, 3.4, 2.1, 4.5, 6.3, 10.7, 8.9, 9.4, 9.4, 7.6, 10.0, 3.3, 6.7, 7.8, 11.6, 13.8 and, 18.6. Assume that the smiling times follow a uniform distribution between zero and 23 seconds, inclusive. Note that zero and 23 are the lower and upper cut-offs for the uniform distribution of smiling times.
Since the smiling times' distribution is a uniform distribution, it can be said that any smiling time from zero to and including 23 seconds has an equal likelihood of occurrence. A histogram that can be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.
For this example, the random variable, x = length, in seconds, of an eight-week-old baby's smile. The notation for the uniform distribution is x ~ U(a, b) where a = the lowest value (lower cut-off) of x and b = the highest value (upper cut-off) of x. For this example, a = 0 and b = 23.
The mean, μ, is calculated using the following equation:
The mean for this distribution is 11.50 seconds. The smile of an eight-week-old baby lasts for an average time of 11.50 seconds.
The standard deviation, σ, is calculated using the formula:
The standard deviation for this example is 6.64 seconds.
This text is adapted from Openstax, Introductory Statistics, Section 5.2 The Uniform Distribution