The Uniform Distribution

The uniform circulation is a consistent probability distribution and also is came to with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or to exclude, of endpoints.

The data in (Figure) space 55 laugh times, in seconds, of an eight-week-old baby.

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 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 15.9 16.3 13.4 17.1 14.5 19 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 3.3 6.7 7.8 11.6 13.8 18.6

The sample average = 11.49 and also the sample conventional deviation = 6.23.

We will certainly assume that the laugh times, in seconds, follow a uniform distribution between zero and also 23 seconds, inclusive. This method that any smiling time from zero to and including 23 seconds is same likely. The histogram that might be created from the sample is an empirical circulation that carefully matches the theoretical uniform distribution.

Let X = length, in seconds, of an eight-week-old baby’s smile.

The notation for the uniform circulation is

X ~ U(a, b) wherein a = the lowest value of x and b = the highest value the x.

The probability density role is f(x) = because that axb.

For this example, X ~ U(0, 23) and also f(x) = because that 0 ≤ X ≤ 23.

Formulas because that the theoretical mean and also standard deviation are and For this problem, the theoretical mean and also standard deviation are

μ = = 11.50 seconds and σ = = 6.64 seconds.

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Notice the the theoretical mean and also standard deviation are close to the sample mean and also standard deviation in this example.