### 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.