CDF from PDF: Let X be random variable w/ density function f(x) = e^{-C|x|}; find C

Mathcatchup

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Let \(\displaystyle X\) be a random variable with density function \(\displaystyle f(x)\, =\, e^{-C\,|x|},\, x\, \in\, \mathbb{R}\)

Find \(\displaystyle C\)



The answer is 2.
However, I am having hard time getting there.

To get the answer I need to:
Integrate f(x) with 0 to 1 limits and set the answer to be equal to 1 and then solve for C. Correct?
I also have no idea how to integrate an absolute value...
 
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To get the answer I need to:
Integrate f(x) with 0 to 1 limits and set the answer to be equal to 1 and then solve for C. Correct?

This is almost correct, but not quite. You're told that \(\displaystyle x \in \mathbb{R}\), so you need to integrate over all real numbers. But, yes, you would want to set this integral equal to 1 and solve for C.

As for how to integrate an absolute value, you can't directly do so. What you can do, however, is break it down into cases. Think about what you know about the absolute value. When is it positive? When is it negative? What happens to \(\displaystyle |x|\) when \(\displaystyle x \ge 0\)? What happens to \(\displaystyle |x|\) when \(\displaystyle x < 0\)? You may find it helpful to manually calculate some absolute values, such as \(\displaystyle |3|\), \(\displaystyle |5|\), \(\displaystyle |-4|\), or \(\displaystyle |-77|\). As a further hint: What is the value of \(\displaystyle -(-2)\)? How does that relate to the value of \(\displaystyle |-2|\)?
 
This is almost correct, but not quite. You're told that \(\displaystyle x \in \mathbb{R}\), so you need to integrate over all real numbers. But, yes, you would want to set this integral equal to 1 and solve for C.

As for how to integrate an absolute value, you can't directly do so. What you can do, however, is break it down into cases. Think about what you know about the absolute value. When is it positive? When is it negative? What happens to \(\displaystyle |x|\) when \(\displaystyle x \ge 0\)? What happens to \(\displaystyle |x|\) when \(\displaystyle x < 0\)? You may find it helpful to manually calculate some absolute values, such as \(\displaystyle |3|\), \(\displaystyle |5|\), \(\displaystyle |-4|\), or \(\displaystyle |-77|\). As a further hint: What is the value of \(\displaystyle -(-2)\)? How does that relate to the value of \(\displaystyle |-2|\)?

So do I use -inf +inf for my integration boundaries?
 
So do I use -inf +inf for my integration boundaries?

Yes.

And since, as has been mentioned, the behavior of the absolute value changes at x=0, you could break that into two integrals, from -inf to 0 and from 0 to inf.
 
I am stuck. I don't know how to evaluate the function for minus and plus infinity :mad:

Have you learned about improper integrals? You don't actually evaluate at infinity; you take the limit as the limit of integration approaches infinity. If you haven't learned about it, you shouldn't be given this problem (in fact, you really shouldn't be studying CDF's); but look here for an explanation.
 
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What am I doing wrong?


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What am I doing wrong?

Your second integral should be of e^{cx}, not e^{-cx}. You forgot that when cx is negative, |cx| is -cx, so -|cx| = cx. This is, in fact, the main reason for splitting into two integrals.

You'll find that the two integrals end up being equal (not opposites), which should be obvious if you sketch a graph of e^-|cx|.
 
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