Integration: Exponential Distribution, probability

KangoSkank

I have a few sample questions for an upcoming exam in front of me here and I haven't go a clue how to do this on.. Could anyone please give a hand?

Example: The exponential distribution.
Suppose we have a radioactive sample, the lenght of time any atom sur-
vives before decaying is a random variable taking values in [0,∞). The
proportion of atoms that survive to time t becomes small exponentially as t
increases:

Pr(T ≥ t) = Ce^(−kt).
Let f be the probability density function for T. Then

∫(from t to infinity)f(x)dx = Pr(T≥ t) = Ce^(−kt).

Using FTC we get −f(t) = −Cke^(−kt), so f(t) = Cke^(−kt).

Question: C =?

Hint: Use that
∫(from 0 to infinity) f(x)dx = 1.

Thanks

TommyGunne

Is this not just 1?

Fremen

yes, you don't even need to find f. You just use that Ce^{-kt} = 1 when t = 0