double exponential exercise!

[federicoforzano]

Hi everybody, I'm a university student from Argentina. I'm learning to speak in english so I apologize if something is not understood.

My teacher sent some exercises to solve but he's not taking the time to explain it. Looking for answers I found this forum and I think you could help me.

Following the rules of the forum, I will make different posts for the exercises that I have.

I tried some things but I'm really lost.

If you could tell me how to solve it or what I could do, I would really appreciate it.

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An anonymous variable X is said to be double exponential with a parameter λ> 0 if and only if its density function is:

1) Determine the value of "c" and find μ = E [X], σ 2 = V [X] and the standard deviation of X, σ.
2) Find the cumulative distribution function of X, that is, F X (x), where x ∈ R.
3) Find the density function f Y (y) of the anonymous variable Y = | X |. What type of known random variable is Y?

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Thanks you again.

 

[Steven G]

In the future please follow the posting guidelines that in part asks you to show your work.
1) [math]\int_{-\infty}^\infty f(x)dx=1[/math]. Solve that integral for c.

Once you have c, then [math]E[X] = \int_{-\infty}^\infty xf(x)dx[/math].

Your English is fine.

 

[federicoforzano]

Sorry for not sharing my work, what I have solved so far is point 1, I share some images with what I did.
Any feedback is welcome.

(a) (b)

(c)

 

[Romsek]

E[X] and thus Var[X] are incorrect.

Think about it a sec before diving into the integrations. You have a distribution symmetric about 0. What is going to be the expected value?