Suppose we select a point X at random from an interval [a, b], where the probability that a point is in any subinterval of [a, b] only depends on the length of the subinterval. X has a uniform(a, b) distribution, with a constant probability density of 1/(b − a) over the interval [a, b] and zero everywhere else.
| Parameter | Range | Description |
|---|---|---|
| a | −∞ < a < b | Lower bound of interval |
| b | a < b < ∞ | Upper bound of interval |
Probability Density Function
Support
Mean
Variance
| Example | a | b |
|---|---|---|
| A number is chosen randomly from the interval [0, 1]. Let X be the value of the number. | 0.000 | 1.000 |
| A spinner is spun randomly until it comes to rest. Let X be the angle (in radians) that the spinner makes to the horizontal axis. | -3.142 | 3.142 |
| You arrive at a subway station for a train that runs every three minutes, but don't know when the previous train left. Let X be the time spent waiting for the train. | 0.000 | 3.000 |
X ∼ Uniform(a, b)
E(X) = , Var(X) =
The uniform distribution is used when an event is known to occur at some point in a bounded interval but no further information is given.
The uniform distribution is used when an event is known to occur at some point in a bounded interval but no further information is given.
The illustration above shows a point X chosen randomly and with uniform probability from the interval [a, b]. If the probability that X lies in any subinterval I is equal to length(I)/(b -a), then X is a uniform(a, b) random variable.
The simulation above shows a point X chosen randomly and with uniform probability from the interval [a, b]. The histogram accumulates the results of each simulation.
Y =
E(Y) = , Var(Y) =