Math Problem Statement

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

Solution

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

Var(X)=(ba)212\text{Var}(X) = \frac{(b - a)^2}{12}

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

Let's calculate the variance:

Var(X)=(3(1))212=(3+1)212=4212=1612=43\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 43\frac{4}{3}.

So, the correct answer is:

c. 43\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][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][-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.

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Math Problem Analysis

Mathematical Concepts

Probability
Uniform Distribution
Variance

Formulas

Variance of Uniform Distribution: Var(X) = (b - a)^2 / 12

Theorems

Variance of Uniform Distribution

Suitable Grade Level

Grades 11-12