Associated cumulative distribution function

Hi everybody, I'm new so don't be harsh also I'm not a native speaker

I'm stuck with an exercise from my statistics class. I hope someone will take a few moments to have a look at it:

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Problem 2 (Exponential Distribution)

Consider the random variable X with density

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}0&x\, <\, 0\\c\, \cdot\, \mbox{exp}(-\lambda x)&x\, \geq\, 0 \end{cases}\)

with \(\displaystyle \, \lambda\, >\, 0.\)

(a) Determine c such that X is a random variable (two properties have to be fulfilled).

(b) Determine the associated cumulative distribution function (cdf) F (x).

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in a) I determined that c > 0 and c must be a real number

in b) I know I have to integrate f(x) to get F(x)s, with the bound of integration -∞ and x

so I integrated by substitution and got the antiderivative -(ce^-λx)/λ

now I have to insert x and -∞ and subtract the latter..

so when I insert x I get the antiderivative that I mentioned above but what happens when I insert -∞ is that the whole thing is going towards -∞ and this kind of doesn't make sense to me

I would be really glad if someone could tell me what I did wrong or how I can write this function down on my exercise sheet.. (I really need this point)

Thanks a lot in advance!

biomensch

biomensch said:

Hi everybody, I'm new so don't be harsh also I'm not a native speaker

I'm stuck with an exercise from my statistics class. I hope someone will take a few moments to have a look at it:

---

Problem 2 (Exponential Distribution)

Consider the random variable X with density

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}0&x\, <\, 0\\c\, \cdot\, \mbox{exp}(-\lambda x)&x\, \geq\, 0 \end{cases}\)

with \(\displaystyle \, \lambda\, >\, 0.\)

(a) Determine c such that X is a random variable (two properties have to be fulfilled).

(b) Determine the associated cumulative distribution function (cdf) F (x).

---

in a) I determined that c > 0 and c must be a real number

in b) I know I have to integrate f(x) to get F(x)s, with the bound of integration -∞ and x

so I integrated by substitution and got the antiderivative -(ce^-λx)/λ

now I have to insert x and -∞ and subtract the latter..

so when I insert x I get the antiderivative that I mentioned above but what happens when I insert -∞ is that the whole thing is going towards -∞ and this kind of doesn't make sense to me

I would be really glad if someone could tell me what I did wrong

Click to expand...

The CDF
\(\displaystyle F(x) = \left\{ \begin{gathered} 0,~x \leqslant 0 \hfill \\ \int_0^x {c{e^{ - \lambda t}}dt} ,~x > 0 \hfill \\
\end{gathered} \right.\)

pka said:

The CDF
\(\displaystyle F(x) = \left\{ \begin{gathered} 0,~x \leqslant 0 \hfill \\ \int_0^x {c{e^{ - \lambda t}}dt} ,~x > 0 \hfill \\
\end{gathered} \right.\)

Click to expand...

Ok thanks, so when I integrate this with the bounds you gave me it's the right answer I think.

Sadly I can't solve the next part of the Exercise either. I would appreciate it a lot if somebody could help me there as well

c) Determine E(X) and Var(X)

I know that E(X) is the integral of x*f(x)
so the integral from -∞ to 0 is 0 + the integral from 0 to ∞ would be E(X)
is this true? If it is I'm on the right way, but I can't manage to calculate this integral. I tried and seeked help online with several integral calculators but they gave me different results.. I tried substitution and partial integration but it didn't work..

Thanks for your help!

biomensch said:

c) Determine E(X) and Var(X)

I know that E(X) is the integral of x*f(x)
so the integral from -∞ to 0 is 0 + the integral from 0 to ∞ would be E(X) CORRECT
, but I can't manage to calculate this integral. I tried and seeked help online with several integral calculators but they gave me different results..

Click to expand...

Take a look at this indefinite intergral.

In terms of E^2(X) and E(X^2) how is Var(X) defined?

pka said:

Take a look at this indefinite intergral.

In terms of E^2(X) and E(X^2) how is Var(X) defined?

Click to expand...

the linked indefinite integral is not the same, but I'm sure you know that.. unluckily I'm not a math pro so I don't see what you want to tell me with that..
http://www.wolframalpha.com/input/?i=integral+xe^(-λx)dx
this would be the one I'm looking for, and although I believe that this result is true, I can't compute it without a calculator.
On the exam I'm not allowed to use one, so I need to know how to compute it without any help.

As far as I know Var(X) =E(X²) - E(X)²

Thanks again, I appreciate it a lot!