Examples with Erf, standard normal CDF, etc.

Integrate[(1/(Sqrt[2π]))(E^(-(1/2)(x^2))),x]

Integrate[(1/(Sqrt[2π]))(E^(-(1/2)(x^2))),{x,-∞,t}]

Define the Cumulative Distribution Function for the standard normal PDF :

Φ[x_]=(1/2) (1+Erf[(x/(Sqrt[2]))])

■ The following formulae are explained in J.D. Blume, Statist. Med. 2002; 21:2563-2599

We have two hypotheses about the mean of data following the gaussian PDF with known variance (σ^2), the two hypotheses differ by an amount Δ. The number of observations is n.
We speak of misleading evidence when the likelihood ratio "strongly" favours the "wrong" hypothesis by a factor of at least k=8.
The probability of observing misleading evidence  [J.D. Blume (2002) - formula (9)] as a function of the number of observations n and of the likelihood ratio k, when the two hypotheses differ by an amount Δ=σ, is:

M[n_,k_]=Φ[-((Sqrt[n])/2)-((Log[k])/(Sqrt[n]))]

The probability of observing weak evidence  [J.D. Blume (2002) - formula (11)] for either hypothesis, i.e. a likelihood ratio between 1/k and k, when Δ=σ, is:

W[n_,k_]=Φ[((Sqrt[n])/2)+((Log[k])/(Sqrt[n]))]-Φ[((Sqrt[n])/2)-((Log[k])/(Sqrt[n]))]

Show M[n, k = 8] (thick line) and W[n, k = 8] (dashed line), when Δ=σ, for the number of observations n going from 0 to 30:

Plot[{M[n,8],W[n,8]},{n,0,30},PlotStyle->{Thick,Dashed},PlotRange->{{-1,31},{-0.01,0.16}},Frame->True]