Statistics:
Uniform Distribution (Continuous)
The uniform distribution (continuous) is one of the simplest probability distributions in statistics. It
is a continuous distribution, this means that it takes values within a specified range, e.g. between 0 and 1.
The probability density function for a uniform distribution taking values in the range ato bis:
f(x) =
1
b−aif a≤x≤b
0otherwise
Example
You arrive into a building and are about to take an elevator to the your floor. Once you call the elevator,
it will take between 0 and 40 seconds to arrive to you. We will assume that the elevator arrives uniformly
between 0 and 40 seconds after you press the button. In this case a= 0 and b= 40.
Calculating Probabilities
Remember, from any continuous probability density function we can calculate probabilities by using
integration.
P(c≤x≤d) = Zd
c
f(x)dx =Zd
c
1
b−adx =d−c
b−a
In our example, to calculate the probability that elevator takes less than 15 seconds to arrive we set
d= 15 and c= 0. The correct probability is 15−0
40−0=15
40 .
Expected Value
The expected value of a uniform distribution is:
E(X) = Zb
a
xf(x)dx =Zb
a
x
b−adx =b−a
2
In our example, the expected value is 40−0
2= 20 seconds.
Variance
The variance of a uniform distribution is:
Var(X) = E(X2)−E2(X)
=Zb
a
x2
b−adx − b−a
2!2
=(b−a)2
12
In our example, the variance is (40−0)2
12 =400
3
Standard Uniform Distribution
The standard uniform distribution is where a= 0 and b= 1 and is common in statistics, especially for
random number generation. Its expected value is 1
2and variance is 1
12
