Calculating expected values for a random variable with a continuous distribution

[Halen]

X is a random variable with a continuous distribution with density f(x)=e^(-2|x|), x e R
How would you calculate E(e^(ax)) for a e R?

Will it be right to take a certain range of a?
And also, can you take the bounds for the integral to be between -Infinity and Infinity?

 

[chiro]

X is a random variable with a continuous distribution with density f(x)=e^(-2|x|), x e R
How would you calculate E(e^(ax)) for a e R?

Will it be right to take a certain range of a?
And also, can you take the bounds for the integral to be between -Infinity and Infinity?

|x| is not integrable over the whole range with a single definition. It is C0 continuous but you can't differentiate it and get a continuous function over the interval of the reals.

If you want to find the expectation break it up into parts where you have a completely integrable function over the associated domain.

As you know |x| = x if x >= 0 and -x if x < 0. That should hopefully give you a head start.

 

[Halen]

Thank you! Helps indeed!
So do you suggest that i solve it separately for the two cases of x?

 

[chiro]

Yes you will have to do that.