The lifetime (in years) of a product is $\displaystyle Y=5X^{0.7}$ where X has an exponential distribution with mean one. Find the distribution function and p.d.f. of Y.
Thank you in advance.
I'd calculate the cdf of Y and then differentiate to get the pdf:
$\displaystyle F(y) = \Pr(Y < y) = \Pr(5 X^{0.7} < y) = \Pr \left( X^{0.7} < \frac{y}{5} \right) = \Pr \left( X < \left( \frac{y}{5} \right)^{10/7} \right)$ $\displaystyle = \int_0^{\left( \frac{y}{5} \right)^{10/7}} g(x) \, dx$
where $\displaystyle g(x)$ is the pdf of X.
Popular question. Asked and answered (in different ways) in the following threads:
http://www.mathhelpforum.com/math-he...tial-dist.html
http://www.mathhelpforum.com/math-he...ction-f-x.html
http://www.mathhelpforum.com/math-he...-lifetime.html
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