Uniform Distribution Reference Cheat Sheet

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 $X \sim U(a,b)$ , the probability density is constant from $a$ to $b$ and zero outside that interval. Probabilities are found using area, so $P(c \le X \le d)=\frac{d-c}{b-a}$ when $a \le c \le d \le b$ . The center of the distribution is the mean $\mu=\frac{a+b}{2}$ , and the spread is measured by $\sigma^2=\frac{(b-a)^2}{12}$ .

The cumulative distribution function increases linearly from $0$ to $1$ across the interval.

Key Facts

For a continuous uniform distribution, write $X \sim U(a,b)$ where $a$ is the minimum value and $b$ is the maximum value.
The probability density function is $f(x)=\frac{1}{b-a}$ for $a \le x \le b$ and $f(x)=0$ otherwise.
The total area under the density curve is $1$ , so $(b-a)\cdot \frac{1}{b-a}=1$ .
For any interval inside the support, $P(c \le X \le d)=\frac{d-c}{b-a}$ when $a \le c \le d \le b$ .
For a continuous distribution, $P(X=c)=0$ for any single exact value $c$ .
The mean and median of $X \sim U(a,b)$ are both $\frac{a+b}{2}$ .
The variance is $\sigma^2=\frac{(b-a)^2}{12}$ and the standard deviation is $\sigma=\frac{b-a}{\sqrt{12}}$ .
The cumulative distribution function is $F(x)=0$ for $x<a$ , $F(x)=\frac{x-a}{b-a}$ for $a \le x \le b$ , and $F(x)=1$ for $x>b$ .

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 $a \le x \le b$ for $X \sim U(a,b)$ .
Probability density function: A function $f(x)$ whose area over an interval gives the probability that $X$ falls in that interval.
Cumulative distribution function: A function $F(x)$ that gives the probability $P(X \le x)$ .
Expected value: The long-run average value of a random variable, equal to $E(X)=\frac{a+b}{2}$ for a uniform distribution.
Variance: A measure of spread around the mean, equal to $\sigma^2=\frac{(b-a)^2}{12}$ 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 $X \sim U(a,b)$ , use interval length times height, not just $\frac{1}{b-a}$ .
Forgetting that $P(X=c)=0$ , 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 $c$ and $d$ are inside $[a,b]$ before applying $P(c \le X \le d)=\frac{d-c}{b-a}$ .
Confusing variance and standard deviation, which is wrong because standard deviation is the square root of variance. For $X \sim U(a,b)$ , $\sigma=\frac{b-a}{\sqrt{12}}$ , not $\frac{(b-a)^2}{12}$ .
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 $X \sim U(2,10)$ . Find $P(4 \le X \le 7)$ .
2 Let $X \sim U(0,24)$ represent the time in hours when an event occurs during a day. Find $P(X>18)$ .
3 For $X \sim U(5,17)$ , calculate the mean $\mu$ , variance $\sigma^2$ , and standard deviation $\sigma$ .
4 Explain why $P(X=6)=0$ but $P(5.9 \le X \le 6.1)$ can be greater than $0$ for a continuous uniform distribution.

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Uniform Distribution Reference Cheat Sheet

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