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.

 

[blue_raver22]

Calculating expected values for a random variable with a continuous distribution involves finding the average value of the random variable over all possible outcomes. This can be done by taking the integral of the random variable multiplied by its probability density function (PDF) over the entire range of the random variable. In this case, the random variable X has a continuous distribution with density f(x) = e^(-2|x|), x e R.

To calculate E(e^(ax)) for a e R, we can use the formula for the expected value of a function of a random variable, which is given by E(g(x)) = ∫g(x)f(x)dx, where f(x) is the PDF of the random variable. In this case, g(x) = e^(ax) and f(x) = e^(-2|x|).

To find the expected value, we can take the integral of e^(ax)e^(-2|x|) over the entire range of X, which is from -Infinity to Infinity. This is because the PDF is defined for all real numbers. However, it is important to note that the integral of e^(-2|x|) over the entire range is actually infinite, so we must take the absolute value of the integral to get a finite result.

Therefore, the calculation for E(e^(ax)) is as follows:

E(e^(ax)) = ∫e^(ax)e^(-2|x|)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx = ∫e^(-2|x|+ax)dx

Since the integral of e^(-2|x|) over the entire range is infinite, we cannot simply take the absolute value of the integral. Instead, we can use a trick to