Uniform Distribution Help

What do these properties mean?

• Fully Defined by its Domain

  This means that the only thing necessary to know in order to define a uniform distribution is its start point and its end point. This means that if you know the start point and end point of a uniform distribution, you know the entire distribution.

• No Random Variable

  The uniform distribution has no random variable because the probability density is known throughout the entire distribution. As long as the variable you are interested in observing is within the distribution's domain, it has the probability density of any other point in the distribution. If the variable you are interested in observing is not within the distribution's domain, it has a probability density of zero.

• Cumulative Density

  The probability density of any point within a uniform distribution is the same as any other point, so it is more useful to observe the cumulative density. The cumulative density is defined as the probability of any of the points within the domain you are interested in being the true value observed.

Example

When dealing with a continuous distribution, if you see or hear the words "equally likely" or "uniform", you are dealing with a uniform distribution.

For example, imagine that you are collecting data about the time you spend waiting for the bus every morning. You arrive at the bus stop at the same time every morning, and from previous experience you know that the bus can take anywhere from 2 minutes to 10 minutes to get to the stop. You have noticed that any time within this period seems to be equally likely for the bus to show up during. This would be a uniform distribution with a = 2 (because that is the minimum amount of time the bus can take) and b = 10 (because that is the maximum amount of time the bus can take). The only thing left would be to determine what block of time you are interested in observing the probability that it will arrive in. The minimum of that amount of time will be x₁ and the maximum of that amount of time will be x₂. Note that the bus will not always arrive within that block of time - there is only some probability that it will.

Requirements

• The upper bound of the distribution must be at least equal to the upper bound of the domain you are interested in observing (b >= x₂).
• The upper bound of the domain that you are interested in observing must be greater than the lower bound of the domain that you are interested in observing (x₂ > x₁).
• The lower bound of the domain that you are interested in observing must be at least equal to the lower bound of the distribution (x₁ >= a).