The idea

Finding μ and σ

Working backwards from given probabilities to find an unknown mean or standard deviation (or both) by standardising to a z-value and solving — including a pair of simultaneous equations when both are unknown.

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Statistics · Statistical distributions

Finding μ and σ

Why it works

Every probability pins down a z

The usual normal question hands you

\mu

and

\sigma

and asks for a probability. Exams love the reverse: "

10\%

of components last under

800

hours — find

\mu

and

\sigma

." Now the parameters are the unknowns, and the probabilities are the clues. The key is that every probability statement pins down a $z$ -value, and

z

is tied to the unknowns by the standardising equation:

z = \frac{x - \mu}{\sigma}

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