As always, I hope this email finds you well and warm today.
I am having an issue understanding the exponential distribution with rate parameter $\lambda$. I am seeing two different pdfs for this distribution; the one in R says $\lambda\exp(-\lambda x)$, while the one in my textbook says $\frac{1}{\lambda}\exp(\frac{-x}{\lambda})$. I understand that the formula in R calculates $\lambda=\frac{1}{rate}$, but I am totally confused as to what I should use. If $P \sim Exponential(4)$, does this mean its pdf is $(4)\exp(-4x)$, or does it mean its pdf is $\frac{1}{4}\exp(\frac{-x}{4})$? It gives different answers in R as well, as the second argument in pexp()is $rate=\frac{1}{\lambda}$.
My question in particular comes from a homework question:
Let $Z \sim Exponential(4)$. Compute $Pr(Z \geq 5)$.
Ans: $\frac{1}{e^{20}}$
Whereas, I did:
$=>\int_5^\infty \frac{1}{4} e^{\frac{x}{4}} dx$
$=>-1\int_5^\infty e^u du$
$=>e^{\frac{5}{4}}$
Or by using R:
> pexp(5, 1/4, lower.tail=F)
I feel like every week there's a new issue that I'm having with this textbook :P as always please help!
--