Functions of one random variable question

**SpiffWilkie** · November 8th 2012, 09:07 AM

I have a problem as follows:
Let X have the pdf f(x)= θ * e^{( - θ x)}, 0 < x < infinity

find the pdf of Y = e^x

I've gone about this the way I normally do for these problems.
I have
G(y) = P(X < ln y) = integral of θ * e^{( - θ x)} dx from 0 to ln y = 1 - y^-^θ
So then I need g(y) = G'(y) of Y? which would be
θ * e^{( - θ (ln y)}??

or am I missing something super-duper obvious?

Thanks

**Siron** · November 8th 2012, 10:22 AM

Note that the random variable $X \sim \mbox{exp}(\theta)$ (exponential distribution) where the distribution function $F$ is given by $F_{X}(x) = 1-e^{-\theta x}$ (for $x \geq 0$ ). We can compute the distribution function of the random variable $Y=e^X$ as follows: $F_{Y}(x) = P(Y \leq x) = P(e^X \leq x) = P(X \leq \ln(x)) = F_{X}[\ln(x)]$ therefore
$F_{Y}(x) = 1-e^{-\theta \ln(x)}$ . We can find the pdf of $Y$ (let's call it $g(x)$ ) by calculating the first derivative of the distribution function, hence $g(x) = \frac{dF_{Y}(x)}{dx} = \frac{d}{dx}\left[1-e^{-\theta \ln(x)}\right] = \left(\frac{\theta}{x}\right)e^{\theta \ln(x)}$

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