# Thread: distribution function and p.d.f.

1. ## distribution function and p.d.f.

The lifetime (in years) of a product is $Y=5X^{0.7}$ where X has an exponential distribution with mean one. Find the distribution function and p.d.f. of Y.

2. Originally Posted by calabrone
The lifetime (in years) of a product is $Y=5X^{0.7}$ where X has an exponential distribution with mean one. Find the distribution function and p.d.f. of Y.

I'd calculate the cdf of Y and then differentiate to get the pdf:

$F(y) = \Pr(Y < y) = \Pr(5 X^{0.7} < y) = \Pr \left( X^{0.7} < \frac{y}{5} \right) = \Pr \left( X < \left( \frac{y}{5} \right)^{10/7} \right)$ $= \int_0^{\left( \frac{y}{5} \right)^{10/7}} g(x) \, dx$

where $g(x)$ is the pdf of X.

3. Originally Posted by calabrone
The lifetime (in years) of a product is $Y=5X^{0.7}$ where X has an exponential distribution with mean one. Find the distribution function and p.d.f. of Y.