Question 1201439: The number of tickets purchased by an individual for Beckham College’s holiday music festival is a uniformly distributed random variable ranging from 6 to 15.
Find the mean and standard deviation of this random variable. (Round your answers to 2 decimal places.)
Answer by math_tutor2020(3817)   (Show Source):
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Answers:
Mean = 10.5
Standard Deviation = 2.87 (approximate)
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Explanation:
The number of tickets is a discrete variable.
It can only be equal to one of the following whole numbers {6,7,8,9,10,11,12,13,14,15}
This random variable would be continuous if it made sense to have some fractional amount of tickets (eg: 6.785 tickets), but of course it doesn't make sense to have a fractional amount of tickets.
The mean of a discrete uniform random variable is found by adding up the endpoints, and then dividing in half.
We are computing the midpoint.
midpoint = (a+b)/2
midpoint = (6+15)/2
midpoint = 21/2
midpoint = 10.5
The mean is 10.5
It represents the center of the distribution.
The proof should be fairly straight-forward, but let me know if you need me to go into more detail in this regard.
The standard deviation of a discrete uniform random variable isn't as straight-forward.
The formula is
One proof is found here
https://proofwiki.org/wiki/Variance_of_Discrete_Uniform_Distribution
Plug in a = 6 and b = 15
The standard deviation is approximately 2.87
The standard deviation is a measure how spread out a distribution is.
Side note:
Some textbooks will present the standard deviation formula as
where n = b-a+1 represents the number of whole numbers from a to b, including both endpoints.
Example: {6,7,8} has n = b-a+1 = 8-6+1 = 3 items.
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