# Finding A. Prob.

Jbreezy

## Homework Statement

The expoential distribution is given by

f(x) = 0 for x <0 and f(x) = Ae^-cx for x >= 0

## The Attempt at a Solution

Find A so that f(x) is a probability distribution. So all that I did was use the def.

I said A∫e^(-cx) dx = 1 from 0 to ∞ I integrated it I got that A/c = 1 so A = c.

So then the next question says find P(x <1) so I did ∫ce^-ct dt = -e^(-ct) between [0,1] so I'm just confused because I have c still it isn't like it's going to drop out. So I'm thinking I did something wrong. Should A be numerical? I don't see how . Thanks

## Answers and Replies

brmath
Probably a statistician would know this, but I don't. What is P(x)?

Jbreezy
Probability. Idk. If they say find the probability that x <1 then you express it as P( x<1)

Mentor

## Homework Statement

The expoential distribution is given by

f(x) = 0 for x <0 and f(x) = Ae^-cx for x >= 0

## The Attempt at a Solution

Find A so that f(x) is a probability distribution. So all that I did was use the def.

I said A∫e^(-cx) dx = 1 from 0 to ∞ I integrated it I got that A/c = 1 so A = c.

So then the next question says find P(x <1) so I did ∫ce^-ct dt = -e^(-ct) between [0,1] so I'm just confused because I have c still it isn't like it's going to drop out. So I'm thinking I did something wrong. Should A be numerical? I don't see how . Thanks

Since you aren't given a value for c, you aren't going to get a value for A that is a specific number. I don't see anything wrong in your work.

Jbreezy
OK. So report my answer for P( x<1 ) = ∫ce^-ct dt = -e^(-ct) between [0,1]
as 1/e^c

Mentor
I'm not sure you're writing what you mean.

$$\int e^{-ct}dt = -\frac 1 c \int e^{-ct} (-c dt) = -\frac 1 c e^{-ct} + C$$

For the definite integral you get 1/c. Note that I haven't included A, but if you multiply throughout by A, you get A/c. For that to be equal to 1, then A = c, which is what you had.

Homework Helper
Gold Member
OK. So report my answer for P( x<1 ) = ∫ce^-ct dt = -e^(-ct) between [0,1]
as 1/e^c

That isn't correct. ##P(X<1)=\int_0^1 ce^{-cx}~dx## and you have left out the lower limit in your calculation.

Also, I think most texts define the exponential density function as ##\lambda e^{-\lambda x}## for ##x > 0## in the first place.

Jbreezy
This was my course pack. What do you mean it isn't right ? What isn't right? my answer for my integral 1/e^c?

I have the lower limit [0,1] what are you talking about.

Jbreezy
OH I see the lower limit missing. But what else isn't right?

Jbreezy
## P(X<1)=\int_0^1 ce^{-cx}~dx = 1-1/e^c ## Is this correct?

And is
## P(.5 < X<1.5)=\int_0^1 ce^{-cx}~dx = 1/e^c(0.5) - 1/e^c(1.5) ## correct?
I don't get what your saying is wrong LCKurtz

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Mentor
## P(X<1)=\int_0^1 ce^{-cx}~dx = 1-1/e^c ## Is this correct?
I think so. It's been a long time since I studied mathematical statistics, and I seem to recall that this is a Poisson distribution.
And is
## P(.5 < X<1.5)=\int_0^1 ce^{-cx}~dx = 1/e^c(0.5) - 1/e^c(1.5) ## correct?
I don't get what your saying is wrong LCKurtz
## P(.5 < X<1.5)=\int_{.5}^{1.5} ce^{-cx}~dx ##

In general, ## P(a < X< b)=\int_a^b ce^{-cx}~dx ##, where a ≥ 0 and b ≥ a.

Homework Helper
Gold Member
## P(X<1)=\int_0^1 ce^{-cx}~dx = 1-1/e^c ## Is this correct?

Yes

And is
## P(.5 < X<1.5)=\color{red}{\int_0^1} ce^{-cx}~dx = 1/e^c(0.5) - 1/e^c(1.5) ## correct?

Put the correct limits on the integral and it will be.

Jbreezy
Oh I'm sorry. I just copied your tex and forgot to change it. It is OK on my paper.
Thanks

Homework Helper
Dearly Missed

## Homework Statement

The expoential distribution is given by

f(x) = 0 for x <0 and f(x) = Ae^-cx for x >= 0

## The Attempt at a Solution

Find A so that f(x) is a probability distribution. So all that I did was use the def.

I said A∫e^(-cx) dx = 1 from 0 to ∞ I integrated it I got that A/c = 1 so A = c.

So then the next question says find P(x <1) so I did ∫ce^-ct dt = -e^(-ct) between [0,1] so I'm just confused because I have c still it isn't like it's going to drop out. So I'm thinking I did something wrong. Should A be numerical? I don't see how . Thanks

The value A = c is correct. Also:
$$P(X \leq v) = 1 - e^{-cv}, \: \: P(X > v) = e^{-cv}.$$
These are used so often that if you plan to do more Probability in the future you should commit them to memory, along with
$$E X = \frac{1}{c}, \; \; \text{Var} X = \frac{1}{c^2}.$$

BTW: the standard way of writing is to use ##e^{-cv}## rather than ##1/e^{cv}##. They are, of course, eqiuivalent, but the first way is used all the time by everybody, at least in works on probability.

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Jbreezy
OH Ok cool thanks for the info ray.