91影视

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

It is given that Z has a standard normal distribution then pdf of Z can be written as:

\[{{\text{f}}_{\text{z}}}{\text{(z) = }}\frac{{\text{1}}}{{\sqrt {{\text{2\pi }}} }}{\text{ \times exp}}\left( {\frac{{{\text{ - }}{{\text{z}}^{\text{2}}}}}{{\text{2}}}} \right)\]

It is also given that Y is a linear function of Z such that:

\[{\text{Y = \sigma Z + \mu }}\]

If we denote the cdf of \[{\text{Z}}\] and Y as F_z(z)and F_y(y) respectively, then we can write cdf of Y as:

\[\begin{gathered}

{{\text{F}}_{\text{y}}}{\text{(y) = P(Y\poundsy)}} \\

{\text{ = P(\sigma Z + \mu \poundsy)}} \\

{\text{ = P}}\left( {{\text{Z\pounds}}\frac{{{\text{y - \mu }}}}{{\text{\sigma }}}} \right){{\text{F}}_{\text{y}}}{\text{(y)}} \\

{\text{ = }}{{\text{F}}_{\text{z}}}\left( {\frac{{{\text{y - \mu }}}}{{\text{\sigma }}}} \right) \\

\end{gathered} \]

We denote the pdf of Y as \[{{\text{f}}_{\text{y}}}{\text{(y)}}\] , then

\[\begin{gathered}

{{\text{f}}_{\text{y}}}{\text{(y) = }}\frac{{\text{d}}}{{{\text{dy}}}}{{\text{F}}_{\text{y}}}{\text{(y)}} \\

{\text{ = }}\frac{{\text{d}}}{{{\text{dy}}}}{{\text{F}}_{\text{z}}}\left( {\frac{{{\text{y - \mu }}}}{{\text{\sigma }}}} \right){{\text{f}}_{\text{y}}}{\text{(y)}} \\

{\text{ = }}\frac{{\text{1}}}{{\text{\sigma }}}{{\text{f}}_{\text{z}}}\left( {\frac{{{\text{y - \mu }}}}{{\text{\sigma }}}} \right) \\

\end{gathered} \]

Using equations (1) and (2), we can write:

\[\begin{gathered}

{{\text{f}}_{\text{y}}}{\text{(y) = }}\frac{{\text{1}}}{{\text{\sigma }}}{\text{ \times }}\frac{{\text{1}}}{{\sqrt {{\text{2\pi }}} }}{\text{ \times exp}}\left( {\frac{{{\text{ - }}{{\left( {\frac{{{\text{y - \mu }}}}{{\text{\sigma }}}} \right)}^{\text{2}}}}}{{\text{2}}}} \right){{\text{f}}_{\text{y}}}{\text{(y)}} \\

{\text{ = \times }}\frac{{\text{1}}}{{{\text{(\sigma )}}\sqrt {{\text{2\pi }}} }}{\text{ \times exp}}\left( {\frac{{{\text{ - [y - \mu }}{{\text{]}}^{\text{2}}}}}{{{\text{2}}{{\text{\sigma }}^{\text{2}}}}}} \right] \\

\end{gathered} \]

From a look at the above equation, we can say that Y is also normally distributed with parameters \[{\text{\mu }}\] and \[{\text{\sigma }}\] .