Question

The number of tickets purchased by an individual for Beckham College's holiday music festival is a uniformly distributed random variable ranging from 1 to 12. Mean? Standard Deviation?

Answer 1

1) Mean: 6.5

2) Standard deviation: 3.2

Step-by-step explanation:

1)

A uniform random distribution is the distribution of a variable that can assume n different values, and each of these values is equally likely to occur.

This means that the probability of occurring of each of these values is the same:

$p(x_1)=p(x_2)=...=p(x_n)$

For a uniform random distribution, the mean of the distribution is simply equal to the median value of the distribution, which is exactly at the center of the distribution itself. It can be calculated as

$\bar x = \frac{1}{2}(a+b)$

where

a is the lowest value of the distribution

b is the highest value of the distribution

Here the distribution consists of all values ranging from 1 to 12, so:

a = 1

b = 12

Therefore the mean value is

$\bar x=\frac{1}{2}(1+12)=6.5$

2)

For a uniform random distribution, it can be dimostrated that the variance of the distribution is

$\sigma^2 = \frac{1}{12}(b-a)^2$

For the distribution in this problem, we have:

a = 1

b = 12

Therefore the variance is

$\sigma^2 = \frac{1}{12}(12-1)^2=10.08$

The standard deviation is the square root of the variance, therefore for this distribution it is:

$\sqrt{\sigma^2}=\sqrt{10.08}=3.2$

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