# Math Help - Transformations of random variables with the exponential distribution

1. ## Transformations of random variables with the exponential distribution

Just posted a similar thread, but I'm not sure if you're allowed multiple questions in one thread (sorry if you are!)

$X$ is exponentially distributed with parameter $\lambda$. For the random variable $Y=\frac{1}{X}$
i) State the range of values $Y$ can take.
ii) Obtain the distribution function of $Y$, $F_{Y}(y)$.
iii) Obtain the probability density function of $Y$, $f_{Y}(y)$.

2. First of all I would like to know how you write your exponential distribution.
And the range is $(0,\infty)$

$F_Y(y)=P(Y\le y)=P(1/X\le y)=P(X\ge 1/y)=\lambda\int_{1/y}^{\infty}e^{-x\lambda} dx$

3. $\lambda e^{-\lambda x}$ is how we use the exponential distribution.