Recipe

Find the capacity for a stated confidence level

Use when: a parameter is uniform on [a, b] and you are given a target confidence level 1 - alpha (or a risk level alpha to convert first)

Goal: the smallest capacity x satisfying P(theta <= x) >= 1 - alpha

Quick recipe

1
Write down the range $[a,b]$ and the confidence level $1-\alpha$ . If only a risk level $\alpha$ is stated, compute $1-\alpha$ first.

Why: the quantile formula below needs exactly these three numbers, in this form.
2
Compute $x=a+(1-\alpha)(b-a)$ .

Example

$[a,b]=[100,200]$ , confidence $0.9$ : $x=100+0.9(100)=190$ .

Why: this is the exact point where the uniform distribution's cumulative probability first reaches $1-\alpha$ .
3
Check $a\le x\le b$ .
- the result falls outside $[a,b]$ — the stated confidence level was outside $[0,1]$ ; recheck the inputs.
- the result falls inside $[a,b]$ — the capacity is confirmed.
Why: a confidence level genuinely between $0\%$ and $100\%$ always produces a capacity inside the parameter's own plausible range.

Common mistakes

Plugging in $\alpha$ where the formula wants $1-\alpha$ , which silently swaps the required capacity with the LEFTOVER risk.
Forgetting that $\alpha=0$ (zero risk) is a real, valid input. It just recovers the full worst-case capacity, $b$ .
Applying this exact formula to a non-uniform distribution, where the quantile is generally NOT this simple linear expression.