How to find the height of a uniform distribution?
Understand the Problem
The question is asking how to calculate the height of a uniform distribution, which involves understanding the concept of uniform probability density functions and how they are defined. The height can typically be found using the formula for the uniform distribution, which is 1 divided by the interval length.
Answer
The height of the uniform distribution is given by $f(x) = \frac{1}{b - a}$.
Answer for screen readers
The height of the uniform distribution is given by the formula:
$$ f(x) = \frac{1}{b - a} $$
where $a$ and $b$ are the limits of the interval.
Steps to Solve
- Identify the interval length
To find the height of a uniform distribution, you first need to know the interval over which the distribution is defined. For example, if the variable $X$ is uniformly distributed between $a$ and $b$, the interval length is calculated as:
$$ L = b - a $$
- Calculate the height
The height of the uniform distribution (i.e., the probability density function, $f(x)$) is defined as the reciprocal of the interval length. It can be found using the formula:
$$ f(x) = \frac{1}{L} $$
Substituting the interval length from the previous step, we get:
$$ f(x) = \frac{1}{b - a} $$
- Specify the range of the distribution
The probability density function is defined for the interval $[a, b]$. You should also note that outside this interval, the height (density) is zero:
$$ f(x) = 0 \text{ for } x < a \text{ or } x > b $$
The height of the uniform distribution is given by the formula:
$$ f(x) = \frac{1}{b - a} $$
where $a$ and $b$ are the limits of the interval.
More Information
In a uniform distribution, every outcome in the interval has the same likelihood. This feature makes uniform distributions a common model in probability and statistics, especially when there is no prior information suggesting any outcomes are more likely than others.
Tips
- Confusing the height with probability: Remember that the height represents the probability density, not the actual probability. The area under the curve for the entire interval equals 1.
- Forgetting to specify the interval: Make sure to specify the range $[a, b]$ clearly, as the probability density is only applicable within this range.
