The Uniform Distribution

Uniform distribution tutorial
A uniform distribution is a probability distribution in which every value between an interval from to is equally likely to be chosen.

The probability that we will obtain a value between x1 and x2 on an interval from to can be found using the formula:

P(obtain value between x1 and x2)  =  (x2 – x1) / (b – a)

Uniform distribution example

The uniform distribution has the following properties:

  • The mean of the distribution is μ = (a + b) / 2
  • The variance of the distribution is σ2 = (b – a)2 / 12
  • The standard deviation of the distribution is σ = √σ2

Let’s walk through some examples to get a better understanding of the uniform distribution.

Examples Using the Uniform Distribution

Example 1: 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 using the formula:

a = 0 minutes

b = 20 minutes

x1 = 0 minutes

x2 = 8 minutes

P(bus shows up in 8 minutes or less)  =  (8 – 0) / (20 – 0)  =  8 / 20  =  0.4

Using the calculator:

Plug the following numbers into the Uniform Distribution Calculator:

Uniform distribution calculator

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

Example 2: A bus shows up at a bus stop every 20 minutes. If you arrive at the bus stop, find the time in which there is a 90% chance that the bus will come by that time.

Solution:

Time = [0.9 * (b – a)]  + a  =  [0.9 * (20 – 0)]  + 0  =  18 minutes

Example 3: 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 using the formula:

a = 15 grams

b = 25 grams

x1 = 17 grams

x2 = 19 grams

P(weighs between 17 and 19 grams)  =  (19 – 17) / (25 – 15)  =  2 / 10  =  0.2

Using the calculator:

Plug the following numbers into the Uniform Distribution Calculator:

Uniform distribution calculator example

The probability that a randomly selected frog weighs between 17 and 19 grams is 0.2. 

Example 4: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. Find the 80th percentile for a frog’s weight.

Solution:

Weight  =  [0.8 * (b – a)]  + a  = [0.8 * (25 – 15)]  + 15  =  23 grams

Example 5: 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 2 and a half hours?

Solution using the formula:

a = 2 and a half hours = 150 minutes

b = 170 minutes

x1 = 120 minutes

x2 = 170 minutes

P(game lasts more than 2 and a half hours)  =  (170 – 150) / (170 – 120)  =  20 / 50  =  0.4

Using the calculator:

Plug the following numbers into the Uniform Distribution Calculator:

Uniform distribution example

Example 6: The length of an NBA game is uniformly distributed between 120 and 170 minutes. Find the length of a game in which only 5% of games last longer.

Solution:

Length =  [0.95 * (b – a)]  + a  = [0.95 * (170 – 120)]  + 120  = 167.5 minutes

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