(b-a)^2/12

Definition

(b-a)^2/12 is a key term in the context of the Uniform Distribution, which describes the variance of a random variable that follows a uniform distribution. It represents the formula for calculating the variance of a uniform distribution, where 'a' and 'b' are the lower and upper bounds of the distribution, respectively.

5 Must Know Facts For Your Next Test

1. The formula $(b-a)^2/12$ represents the variance of a random variable that follows a uniform distribution.
2. The variance measures how spread out the values of the random variable are from the mean or average value.
3. The range of the uniform distribution, $(b-a)$, is a key factor in determining the variance.
4. Dividing the squared range by 12 gives the variance of the uniform distribution.
5. The variance of a uniform distribution is independent of the specific values of 'a' and 'b', but rather depends on the difference between them.

Review Questions

• Explain the relationship between the range of a uniform distribution and its variance.
  • The variance of a uniform distribution is directly proportional to the square of the range, or $(b-a)^2$. This is because the uniform distribution has a constant probability density function between the lower bound 'a' and the upper bound 'b', and the spread of the values is determined by the difference between these bounds. Dividing the squared range by 12 gives the final formula for the variance, $(b-a)^2/12$, which captures how the width of the distribution affects the spread of the random variable.
• Describe how the formula $(b-a)^2/12$ can be used to calculate the variance of a uniform distribution.
  • The formula $(b-a)^2/12$ provides a way to calculate the variance of a random variable that follows a uniform distribution. The 'a' and 'b' parameters represent the lower and upper bounds of the distribution, respectively. By plugging these values into the formula and performing the calculation, you can determine the variance of the uniform distribution. This variance measure is an important characteristic that describes how spread out the possible values of the random variable are, which is crucial for understanding the behavior and properties of the uniform distribution.
• Analyze how changes in the range of a uniform distribution (the difference between 'b' and 'a') would impact the value of the variance calculated using the formula $(b-a)^2/12$.
  • The formula $(b-a)^2/12$ shows that the variance of a uniform distribution is directly proportional to the square of the range, or $(b-a)^2$. This means that as the difference between the upper bound 'b' and the lower bound 'a' increases, the variance will also increase at a quadratic rate. Conversely, if the range of the uniform distribution decreases, the variance calculated using the formula will also decrease. Understanding this relationship between the range and variance is crucial for analyzing and interpreting the properties of a uniform distribution, as the variance is a key measure of the spread and dispersion of the random variable.

Related terms

Uniform Distribution:

A continuous probability distribution where the random variable can take on values between a lower bound 'a' and an upper bound 'b' with equal probability.

Variance:

A measure of the spread or dispersion of a random variable, describing how far the values tend to be from the mean.

Random Variable:

A variable that can take on different values with certain probabilities, used to model uncertain or random phenomena.