Standard-normal CDF, tail, and quantile identities for power calculations

Given: AC, the standard normal law $N(0,1)$, and its cumulative distribution function $\mathrm{\Phi }$.

The standard-normal law is constructed under AC, and AC implies countable choice. The Axiom of Choice Standard normal and normal laws The Axiom of Choice implies countable choice

The standard-normal law is the probability measure with density $g(t)=e^{-t^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$; this density is positive, Borel, and has integral one. Standard normal and normal laws The standard normal density has total mass one

The function $g$ is continuous because the exponential is smooth, $t\mapsto -t^2\mathrm{/}2$ is continuous, and a composite and scalar multiple of continuous functions are continuous. The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$ Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs

A continuous real-valued function on a nonempty compact interval attains finite maximum and minimum values. Extreme value theorem: a continuous real function on a nonempty compact subset of $\mathbb{R}$ attains a greatest and a least value

Probability is continuous on increasing event sequences and on decreasing event sequences, the latter because all probabilities are finite. Continuity from below for measures Continuity from above when one set has finite measure

A continuous function on a closed interval takes every value between its endpoint values. Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$

Under countable choice, a singleton is a zero-width box of Lebesgue measure zero, and the integral of a nonnegative measurable density over a null set is zero. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included A nonnegative integral over a null set vanishes

Substitution $u=-t$ in a convergent improper integral is valid for the monotone differentiable bijection $t\mapsto -t$; its derivative reverses orientation. Change of variable in an improper integral

If $A\subseteq B$, then $P(B\setminus A)=P(B)-P(A)$, and $P(A^c)=1-P(A)$. Basic identities for a probability measure

The CDF is $\mathrm{\Phi }(x)=P(Z\le x)$. Cumulative distribution function of a real random variable