A uniform probability density function (PDF) is characterized by a constant height across its range. Unlike other distributions where probabilities may vary, the uniform PDF maintains a steady value from the minimum to the maximum bounds of the random variable it represents. This constancy is what makes the uniform distribution unique.
To better understand how to determine the height of this function, let’s consider a uniform distribution defined over an interval from a to b. The height (often denoted as h) of the uniform PDF can be calculated using the formula:
h = 1 / (b - a)Here’s the breakdown of the formula:
- b: This represents the upper limit of the interval.
- a: This represents the lower limit of the interval.
- b – a: This measures the range or width of the distribution. It indicates how spread out the values are between the lower and upper bounds.
- 1 / (b – a): By taking the reciprocal of the range, we ensure that the total area under the PDF remains equal to 1, which is a fundamental requirement for all probability density functions.
Since the area under the curve of any probability density function must equal 1, using this formula guarantees that the uniform distribution’s height will consistently adjust to maintain this property. Regardless of how wide or narrow the interval is, the uniform PDF integrates to 1 when evaluated over its limits.
For instance, if you have a uniform distribution from 2 to 6, the height would be:
h = 1 / (6 - 2) = 1 / 4 = 0.25This means the probability density function will reach a height of 0.25 across the range from 2 to 6. In summary, the height of a uniform PDF directly depends on the width of the interval it covers, allowing it to fulfill the essential condition of total probability.