# Thread: probability with density functions

1. ## probability with density functions

A radioactive material emits alpha particles at a rate described by the density function
f(t) = .1e^(-.1t)
Find the probability that a particle is emitted in the first 10 seconds, given...
a) NO particle is emitted in the first second
b) NO particle is emitted in the first 5 seconds
c) a particle is emitted in the first 3 seconds
d) a particle is emitted in the first 20 seconds

for C and D, i think i'm supposed to integrate f(t) from 0 to 3 (and 0 to 20), which gives the probability of those events happening. Then to find f(t given the event), would it be f(t) / the probabilities of the above? But then, if a particle is emitted in the first 3 seconds, wouldn't the probability be 1? And I'm not sure how to set up the integral at this point for part D (the 20 seconds), because I'm unsure of the bounds.

I have no idea where to begin with A and B, a little push in the right direction would be great! Thanks

2. If it were me, I'd do the five integrations and then think about it.

$\displaystyle \int_{0}^{10}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.632$

$\displaystyle \int_{0}^{1}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.095$

$\displaystyle \int_{0}^{5}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.393$

$\displaystyle \int_{0}^{3}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.259$

$\displaystyle \int_{0}^{20}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.865$

Now,

1) What do you know about Conditional Distributions?

2) What property of the Exponential Distibution might help us out on this one?

3. The probability of event A, given event B, is the probability of A intersect B over the probability of B...in terms of the PDF, f(t|a) = f(t)/p(a)?

4. Okay. Now do it.

5. I have no idea as to what the answer to #2 would be.
I'm pretty sure I figured out parts a and b...

6. So for a and b, i did 1-integral (from 0 to 1, and 0 to 5) then divided f(t) by these values and integrated from 1 to 10 and 5 to 10, i think this is correct.

However, when I apply this method to c (except not doing 1-integral, since the integral is the probability of the event itself) I keep getting numbers larger than 1, which I know is incorrect