A random variable has uniform distribution over the interval [-1,3]. This distribution has variance equal to

a.

13/4

b.

9/2

c.

4/3

d.

8/5

For a random variable \(X\) uniformly distributed over the interval \([a, b]\), the variance \(\text{Var}(X)\) is given by the formula:

\[
\text{Var}(X) = \frac{(b - a)^2}{12}
\]

Here, the interval is \([-1, 3]\), so \(a = -1\) and \(b = 3\).

Let's calculate the variance:

\[
\text{Var}(X) = \frac{(3 - (-1))^2}{12} = \frac{(3 + 1)^2}{12} = \frac{4^2}{12} = \frac{16}{12} = \frac{4}{3}
\]

Thus, the variance is \(\frac{4}{3}\).

So, the correct answer is:

**c. \(\frac{4}{3}\)**

---

Would you like further explanation or have any other questions?

Here are 5 follow-up questions related to this topic:

1. What is the mean of a uniform distribution over an interval \([a, b]\)?
2. How do you calculate the probability density function (PDF) for a uniform distribution?
3. What is the standard deviation of a uniform distribution?
4. Can you compute the expected value of a uniform distribution over the interval \([-1, 3]\)?
5. How does the variance of a uniform distribution change if the interval length is doubled?

**Tip:** For a uniform distribution, the variance and the standard deviation are directly related—variance is the square of the standard deviation.

A random variable has uniform distribution over the interval [-1,3]. This distribution has variance equal to

Learn how to calculate the variance of a uniform distribution over the interval [-1, 3] using the formula Var(X) = (b - a)^2 / 12. This example illustrates the process of determining variance for continuous uniform distributions and helps students understand key concepts in probability theory. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation