The continuous uniform distribution is such that the random variable X takes values between α (lower limit) and β (upper limit). In the field of statistics, α and β are known as the parameters of the continuous uniform distribution. We cannot have an outcome either less than α or greater than β.

The probability density function for this type of distribution is:

X ~U (α, β) is the most commonly used shorthand notation read as “the random variable x has a continuous uniform distribution with parameters α and β”

The total probability (1) is spread **uniformly** between the two limits. **Intervals of the same length have the same probability.**

3 Properties of the Continuous Uniform Distribution:

- For all α ≤ x
_{1}< x_{2}≤ β - P(X < α or X > β = 0)
- P(x
_{1 }≤ X ≤ x_{2}) = (x_{2 }– x_{1})/(β – α)

**Moments**

Mean = (α + β)/2

Variance = (β – α)^{2}/12

**Example**

You have been given that Y~U(100,300). Calculate P(Y > 174) and P(100 < Y < 226

The probability density function is given by:

f_{x}(x) = 1/(300 – 100) = 1/200

Therefore, each “unit interval” has a probability of 1/200.

Which means that P(Y > 174) = (300 – 174)/ 200 = 126/200 = 0.63

Similarly, P(100 < Y < 226) = 0.63 because the interval has the same length as above (126) hence the same probability.

The cumulative distribution function of the continuous uniform distribution looks like this:

The CDF is linear over the variable’s range.

**Question**

A random variable X is uniformly distributed between 32 and 42. Calculate the probability that X will be between 32 and 40

A.80%

B. 8

C. 10%

SolutionThe correct answer is A.

First, you should determine the pdf:

f

_{x}(x) = 1/(42 – 32)= 1/10

Therefore, P(32 < Y < 40) = (40 – 32)/10 = 0.8 or 80%.

*Reading 10 LOS 10h:*

*Define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution.*