A uniform distribution models outcomes that are evenly spread across a fixed interval. This cheat sheet helps students recognize when every value in an interval is equally likely and how to calculate probabilities from interval lengths. It is useful for probability density functions, cumulative distribution functions, expected value, variance, and graph interpretation.
Students need these tools to connect geometric area with probability in continuous random variables.
For a continuous uniform random variable , the probability density is constant from to and zero outside that interval. Probabilities are found using area, so when . The center of the distribution is the mean , and the spread is measured by .
The cumulative distribution function increases linearly from to across the interval.
Key Facts
- For a continuous uniform distribution, write where is the minimum value and is the maximum value.
- The probability density function is for and otherwise.
- The total area under the density curve is , so .
- For any interval inside the support, when .
- For a continuous distribution, for any single exact value .
- The mean and median of are both .
- The variance is and the standard deviation is .
- The cumulative distribution function is for , for , and for .
Vocabulary
- Uniform distribution
- A probability distribution where all values in a given interval are equally likely.
- Support
- The set of values where a random variable can have nonzero probability density, such as for .
- Probability density function
- A function whose area over an interval gives the probability that falls in that interval.
- Cumulative distribution function
- A function that gives the probability .
- Expected value
- The long-run average value of a random variable, equal to for a uniform distribution.
- Variance
- A measure of spread around the mean, equal to for a uniform distribution.
Common Mistakes to Avoid
- Using the height as the probability, which is wrong because probability is area under the density curve. For , use interval length times height, not just .
- Forgetting that , which is wrong for continuous random variables because a single point has no width and no area.
- Using the interval endpoints incorrectly, which can give impossible probabilities. Make sure and are inside before applying .
- Confusing variance and standard deviation, which is wrong because standard deviation is the square root of variance. For , , not .
- Treating the continuous uniform graph like a histogram with separate bars, which is wrong because the density is a smooth constant height across the whole interval.
Practice Questions
- 1 Let . Find .
- 2 Let represent the time in hours when an event occurs during a day. Find .
- 3 For , calculate the mean , variance , and standard deviation .
- 4 Explain why but can be greater than for a continuous uniform distribution.
Understanding Uniform Distribution Reference
A uniform model is appropriate only when there is a good reason to treat equal-sized parts of a range as equally likely. A computer random number generator is a common example. It may produce a value from zero up to one, with no part of that range favoured.
Choosing a random time during a period can sometimes fit too, provided events are not more common at certain times. Real measurements often do not fit this model.
Heights, test scores, travel times, and manufacturing measurements usually cluster around some values. Calling data uniform just because its values lie between a minimum and maximum is a common mistake.
The graph requires careful reading. Its height is not the probability of one exact result. For continuous variables, an exact point has no width, so it has no area.
The height tells students how much probability is packed into each unit of horizontal distance. A narrow interval can have a small probability even when the density graph looks tall. Units matter here.
If a random waiting time is measured in minutes, the density has units of probability per minute. Changing from minutes to seconds changes the numerical height of the graph, even though the actual chance of waiting within a physical time interval stays the same.
The cumulative graph gives a different kind of information. It records the chance of being at or below each input value. Its straight rising section has a constant slope because equal horizontal steps add equal amounts of probability.
Students can use cumulative values to find probabilities between two points by subtracting the earlier accumulated chance from the later one. This works even when an interval touches an endpoint. The graph is flat before the possible range because no outcomes have occurred yet.
It is flat after the range because all probability has already accumulated. Confusing the density graph with the cumulative graph leads to incorrect answers, since one graph shows height while the other shows total probability collected so far.
The midpoint is useful for checking whether an answer makes sense. The distribution is balanced around that point, so intervals of the same length placed symmetrically on either side have equal probabilities. Its spread depends on the full width of the range.
If the range becomes twice as wide, the standard deviation becomes twice as large, while the variance becomes four times as large. This difference matters when comparing formulas. A uniform random variable can be rescaled for practical work.
For example, a value chosen uniformly from zero to one can be converted to a value in any desired range by multiplying by that range width and then shifting it to the correct starting value. This idea appears in simulations, random sampling, and computer graphics. When solving problems, first mark the possible range, clip any requested interval to that range, then use only the part that can actually occur.