91Ó°ÊÓ

Probability: Probability means possibility. It is a branch of mathematics which deals with the occurrence of a random event.

Considering the given information:

Here,\({{\rm{z}}_{{\rm{1 - p}}}}\)denotes the\({{\rm{(100p)}}^{{\rm{th}}}}\) percentile of the Normal distribution with mean\({\rm{\mu }}\) and standard deviation\({\rm{\sigma }}\).

The aim is to find that\({{\rm{(100p)}}^{{\rm{th}}}}\)percentile of the Normal distribution with mean\({\rm{\mu }}\) and standard deviation\(\sigma \) is \({\rm{\mu + }}\left( {\begin{array}{*{20}{l}}{{{{\rm{(100p)}}}^{{\rm{th}}}}{\rm{ for }}}\\{{\rm{ standard normal }}}\end{array}} \right){\rm{.\sigma }}\).

That is to say,

Normal distribution percentiles:

\(\begin{array}{l}\left\{ {\begin{array}{*{20}{l}}{{{{\rm{(100p)}}}^{{\rm{th}}}}{\rm{ percentile }}}\\{{\rm{ for (\mu ,\sigma )}}}\end{array}} \right\}{\rm{ = \mu + }}\left( {\begin{array}{*{20}{l}}{{{{\rm{(100p)}}}^{{\rm{th}}}}{\rm{ for }}}\\{{\rm{ standard normal }}}\end{array}} \right){\rm{.\sigma }}\\{\rm{(or)}}\\\left\{ {\begin{array}{*{20}{l}}{{{{\rm{(100p)}}}^{{\rm{th}}}}{\rm{ percentile }}}\\{{\rm{ for (\mu ,\sigma )}}}\end{array}} \right\}{\rm{ = \mu + }}{{\rm{z}}_{{\rm{1 - p}}}}{\rm{ \times \sigma }}\end{array}\)

Proof:

\(\begin{array}{c}P\left( {X \le \mu + {z_{1 - p}}.\sigma } \right) = P\left( {X - \mu \le {z_{1 - p}}.\sigma } \right)\\ = P\left( {\frac{{X - \mu }}{\sigma } \le {z_{1 - p}}} \right)\\ = P\left( {Z \le {z_{1 - p}}} \right)\end{array}\)

Where\({\rm{Z = }}\frac{{{\rm{X - \mu }}}}{{\rm{\sigma }}}\)

Based on the definition,\({{\rm{z}}_{{\rm{1 - p}}}}\)denotes the\({{\rm{(100p)}}^{{\rm{th}}}}\) percentile of the Normal distribution as\({\rm{\mu + }}{{\rm{z}}_{{\rm{1 - p}}}}{\rm{ \times \sigma }}\).

Therefore, the connection between the general normal percentile and the corresponding z percentile has been established.