What is the probability density function formula for a continuous uniform distribution on the interval [a, b]?

Conditions

• The distribution is continuous and uniform.
• The random variable is defined on the closed interval [a, b].
• The lower bound a is strictly less than the upper bound b.

Reasoning, step by step

1. Identify the lower bound a and the upper bound b of the distribution interval.
2. Calculate the length of the interval by subtracting the lower bound from the upper bound: $b - a$.
3. Divide the total probability (1) by the interval length to find the constant density value: $f(x) = \frac{1}{b - a}$.

Example

For a bus arriving every 60 minutes where the waiting time is uniformly distributed between 0 and 60 minutes, the lower bound is $a = 0$ and the upper bound is $b = 60$. The PDF is calculated as $f(x) = \frac{1}{60 - 0} = \frac{1}{60}$.

Common misconceptions

• Assuming the probability density function represents the probability of a single exact outcome, rather than a density over an interval.
• Forgetting that the function is only non-zero within the interval [a, b] and is zero elsewhere.

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