\( f(x;p) = p \phi(x) (\Phi(-x))^{p-1} \hspace{.3in} x, p > 0 \)

where p is the shape parameter (also referred to as the power parameter), \(\Phi\) is the cumulative distribution function of the standard normal distribution, and \(\phi\) is the probability density function of the standard normal distribution.

As with other probability distributions, the power normal distribution can be transformed with a location parameter, μ, and a scale parameter, σ. We omit the equation for the general form of the power normal distribution. Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

The following is the plot of the power normal probability density function with four values of p.

\( F(x;p) = 1 - (\Phi(-x))^{p} \hspace{.3in} x, p > 0 \)

where \(\Phi\) is the cumulative distribution function of the standard normal distribution.

The following is the plot of the power normal cumulative distribution function with the same values of p as the pdf plots above.

\( G(f) = \Phi^{-1}(1 - (1 - f)^{1/p}) \hspace{.3in} 0 < f < 1; p > 0 \)

where \(\phi^{-1}\) is the percent point function of the standard normal distribution.

The following is the plot of the power normal percent point function with the same values of p as the pdf plots above.

\( h(x;p) = \frac{p\phi(x)} {\Phi(-x)} \hspace{.3in} x, p > 0 \)

The following is the plot of the power normal hazard function with the same values of p as the pdf plots above.

\( H(x,p) = -\log{((\Phi(-x))^{p})} \hspace{.3in} x, p > 0 \)

The following is the plot of the power normal cumulative hazard function with the same values of p as the pdf plots above.

\( S(x;p) = (\Phi(-x))^{p} \hspace{.3in} x, p > 0 \)

The following is the plot of the power normal survival function with the same values of p as the pdf plots above.

\( Z(f) = \Phi^{-1}(1 - f^{1/p}) \hspace{.3in} 0 < f < 1; p > 0 \)

The following is the plot of the power normal inverse survival function with the same values of p as the pdf plots above.