Continuous Uniform Distribution

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:

pdf-continuous-distribution

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:

  1. For all α ≤ x1 < x2 ≤ β
  2. P(X < α or X > β = 0)
  3. P(x1 ≤ X ≤ x2) = (x2 – x1)/(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:

fx(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:

cumulative-distribution-uniform

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%

Solution

The correct answer is A.

First, you should determine the pdf:

fx(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.

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