*54*

AÂ uniform distributionÂ is a probability distribution in which every value between an interval fromÂ *aÂ *toÂ *bÂ *is equally likely to be chosen.

The probability that we will obtain a value between x_{1}Â and x_{2}Â on an interval fromÂ *aÂ *toÂ *bÂ *can be found using the formula:

P(obtain value between x_{1}Â and x_{2})Â =Â (x_{2}Â â€“ x_{1}) / (b â€“ a)

The uniform distribution has the following properties:

- The mean of the distribution isÂ
**Î¼**Â = (a + b) / 2 - The variance of the distribution isÂ
**Ïƒ**= (b â€“ a)^{2}Â^{2}Â / 12 - The standard deviation of the distribution isÂ
**Ïƒ**Â =Â âˆšÏƒ^{2}

**Uniform Distribution in R: Syntax**

The two built-in functions in R weâ€™ll use to answer questions using the uniform distribution are:

**dunif(x, min, max)Â **â€“ calculates the probability density function (pdf) for the uniform distribution whereÂ *xÂ *is the value of a random variable, andÂ *minÂ *and *maxÂ *are the minimum and maximum numbers for the distribution, respectively.Â

**punif(x, min, max)Â **â€“ calculates the cumulative distribution function (cdf) for the uniform distribution whereÂ *xÂ *is the value of a random variable, andÂ *minÂ *and *maxÂ *are the minimum and maximum numbers for the distribution, respectively.Â

*Find the full R documentation for the uniform distribution here.*

**Solving Problems Using the Uniform Distribution in R**

**Example 1:Â ***A bus shows up at a bus stop every 20 minutes. If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less?*

**Solution:**Â Since we want to know the probability that the bus will show up in 8 minutes or less, we can simply use the punif() function since we want to know the cumulative probability that the bus will show up in 8 minute or less, given the minimum time is 0 minutes and the maximum time is 20 minutes:

punif(8, min=0, max=20)

## [1] 0.4

The probability that the bus shows up in 8 minutes or less is **0.4**.

**Example 2:***The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. If you randomly select a frog, what is the probability that the frog weighs between 17 and 19 grams?*

**Solution:**Â To find the solution, we will calculate the cumulative probability of a frog weighing less than 19 pounds, then subtract the cumulative probability of a frog weighing less than 17 pounds using the following syntax:

punif(19, 15, 25) - punif(17, 15, 25)

## [1] 0.2

Thus, the probability that the frog weighs between 17 and 19 grams is** 0.2**.

**Example 3:Â ***The length of an NBA game is uniformly distributed between 120 and 170 minutes. What is the probability that a randomly selected NBA game lasts more than 150 minutes?*

**Solution:**Â To answer this question, we can use the formula 1 â€“ (probability that the game lasts less than 150 minutes). This is given by:

1 - punif(150, 120, 170)

## [1] 0.4

The probability that a randomly selected NBA game lasts more than 150 minutes isÂ **0.4**.