Continuous Uniform Distributions: Parameters and Probabilities

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What is a continuous uniform distribution?

A distribution with constant probability density across a specified interval. Equal-length subintervals within the range have equal probabilities.

What do a and b represent in $X \sim U(a,b)$?

The value a is the lower endpoint, or smallest value, and b is the upper endpoint, or largest value, of the modeled interval.

What is the pdf of $X \sim U(a,b)$ within its interval?

$f(x)=1/(b-a)$ for $a \le x \le b$. This constant height makes the total rectangular area under the density equal to 1.

How do you calculate the theoretical mean of a uniform distribution?

Take the midpoint of the interval: $\mu=(a+b)/2$.

How do you calculate the theoretical standard deviation of a uniform distribution?

Divide the interval width by the square root of 12: $\sigma=(b-a)/\sqrt{12}$.

Quiz questions

1. What does the notation $X \sim U(a,b)$ indicate?Answer: The continuous random variable $X$ has a uniform distribution from $a$ to $b$.
2. What is the probability density of $X \sim U(a,b)$ within its support?Answer: $f(x)=1/(b-a)$ for $a \le x \le b$.
3. For $X \sim U(0,23)$, what is the theoretical mean?Answer: 11.5
4. For $X \sim U(0,23)$, approximately what is the theoretical standard deviation?Answer: 6.64
5. A bus waiting time is uniformly distributed from 0 to 15 minutes. What is the probability that the wait is strictly between 5 and 9 minutes?Answer: $4/15$