Apply transformations

Given a random variable \(Y\) that is a function of a random variable \(X\), \(Y=u(X)\), we can use the distribution function technique to find the pdf of \(Y\).

$$ \begin{align*}
& G\left(y \right)=P \left(Y ≤ y \right)=P\left[u\left(X \right) ≤ y \right] \\
& g \left(y \right)=G^{‘} \left(y \right) \\
\end{align*}
$$

Example

Given the following probability density function of a continuous random variable:

$$ f\left( x \right) =\begin{cases} { x }^{ 2 }+{ 2 }/{ 3, } & 0< x < 1 \\ 0, & otherwise \end{cases} $$

Let \(Y=X^2\). Find \(g(y)\).

$$ \begin{align*}
G\left(y \right) & =P\left(Y ≤ y \right)=P\left[ u \left(X \right) ≤ y \right]=P\left(X^2=Y \right) \\
& =\int _{ 0 }^{ { y }^{ 5 } }{ \left( { t }^{ 2 }+\frac { 2 }{ 3 } \right) dt } =\left( \frac { 1 }{ 3 } \right) { y }^{ \frac { 3 }{ 2 } }+\left( \frac { 2 }{ 3 } \right) { y }^{ \frac { 1 }{ 2 } } \\
g\left( y \right) & ={ G }^{ ‘ }\left( y \right) =\left( \frac { 1 }{ 2 } \right) { y }^{ \frac { 1 }{ 2 } }+\left( \frac { 1 }{ 3 } \right) { y }^{ -\frac { 1 }{ 2 } } \\
\end{align*}
$$

We can also use the change-of-variable technique to find the pdf of a transformed variable, \(Y\).

$$ g\left( y \right) =f\left[ \upsilon \left( y \right) \right] |{ \upsilon }^{ \prime }\left( y \right) | $$

where \(\upsilon \left( y \right)\) is the inverse function of \(y\).

Example

Given the following probability density function of a continuous random variable:

$$ f\left( x \right) =\begin{cases} { x }^{ 2 }+{ 2 }/{ 3, } & 0< x < 1 \\ 0, & otherwise \end{cases} $$

Let \(Y = X – 50\). Find \(g(y)\).

\(v\left(y \right) = Y + 50\)

\(v^{’} \left(y \right) = 1\)

\(g\left( y \right) =f\left[ \upsilon \left( y \right) \right] \left[ { \upsilon }^{ ‘ }\left( y \right) \right] =\left[ { \left( Y+50 \right) }^{ 2 }+\frac { 2 }{ 3 } \right] \left[ 1 \right] ={ \left( Y+50 \right) }^{ 2 }+\frac { 2 }{ 3 } \)

The pdf of a discrete transformed random variable can be found in a similar way, as shown in the formula below:

$$ g\left( y \right) =f\left[ \upsilon \left( y \right) \right] $$

Note: \(|{ \upsilon }^{ ‘ }\left( y \right) |\) is not needed in this case

 

Learning Outcome

Topic 2g: Univariate Random Variables – Apply transformations.


X