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 ≤ x1 < x2 ≤ β
- P(X < α or X > β = 0)
- P(x1 ≤ X ≤ x2) = (x2 – x1)/(β – α)
Mean = (α + β)/2
Variance = (β – α)2/12
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:
The CDF is linear over the variable’s range.
A random variable X is uniformly distributed between 32 and 42. Calculate the probability that X will be between 32 and 40
The correct answer is A.
First, you should determine the pdf:
fx(x) = 1/(42 – 32)
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.