The number of tickets purchased by an individual for Beckham College’s holiday music festival is a uniformly distributed random variable ranging from 4 to 14.

Find the mean and standard deviation of this random variable.

Note: Round your answers to 2 decimal places.



To solve this problem, we'll use the formulas for the mean and standard deviation of a uniformly distributed random variable.

A uniform distribution is defined over an interval \([a, b]\), where every outcome within this interval has an equal likelihood. In this case, the number of tickets \(X\) is uniformly distributed between 4 and 14, meaning \(a = 4\) and \(b = 14\).

### 1. **Mean of a Uniform Distribution:**
The formula for the mean \(\mu\) of a uniformly distributed random variable is:
\[
\mu = \frac{a + b}{2}
\]
Substitute \(a = 4\) and \(b = 14\):
\[
\mu = \frac{4 + 14}{2} = \frac{18}{2} = 9
\]

### 2. **Standard Deviation of a Uniform Distribution:**
The formula for the standard deviation \(\sigma\) of a uniformly distributed random variable is:
\[
\sigma = \sqrt{\frac{(b - a)^2}{12}}
\]
Substitute \(a = 4\) and \(b = 14\):
\[
\sigma = \sqrt{\frac{(14 - 4)^2}{12}} = \sqrt{\frac{10^2}{12}} = \sqrt{\frac{100}{12}} = \sqrt{8.33} \approx 2.89
\]

### Final Answers:
- **Mean**: 9.00
- **Standard Deviation**: 2.89

Would you like any additional details or further explanation?

### Related Questions:
1. How do you compute the probability of selecting a specific range of tickets in a uniform distribution?
2. What is the variance of a uniform distribution?
3. How does the uniform distribution differ from a normal distribution?
4. What are some real-life examples of uniform distributions?
5. How do you calculate the cumulative distribution function (CDF) of a uniform random variable?

### Tip:
When solving problems with uniform distributions, remember that all outcomes within the specified interval are equally likely, which simplifies many calculations.

The number of tickets purchased by an individual for Beckham College’s holiday music festival is a uniformly distributed random variable ranging from 4 to 14. Find the mean and standard deviation of this random variable. Note: Round your answers to 2 decimal places.

Discover how to calculate the mean and standard deviation for a uniform distribution ranging from 4 to 14. Learn the key formulas and concepts behind this statistical problem. Suitable for high school students in Grades 10-12.

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