What is the process for finding the constant A in a probability distribution?

[chris_avfc]

Homework Statement

Probability distribution for finding an object that can move anywhere along an x-axis is given by

P(x) = A x^2 exp(-x^2/a^2)

The Attempt at a Solution

I need to find A so it correctly represents a probability distribution.
Am I right in thinking I need to integrate it?

∫P(x) dx = 1
With the limits of ∞ and 0?

 

[kai_sikorski]

Homework Statement

Probability distribution for finding an object that can move anywhere along an x-axis is given by

P(x) = A x^2 exp(-x^2/a^2)

The Attempt at a Solution

I need to find A so it correctly represents a probability distribution.
Am I right in thinking I need to integrate it?

∫P(x) dx = 1
With the limits of ∞ and 0?

If the object can move anywhere on the x axis, then you need to integrate from -∞ to ∞. Or since the function is even you could use the limits you used but say

∫P(x) dx = 1/2

 

[chris_avfc]

If the object can move anywhere on the x axis, then you need to integrate from -∞ to ∞. Or since the function is even you could use the limits you used but say

∫P(x) dx = 1/2

that makes sense, thank you.

How do I actually integrate the

e^(-x^2/a^2)

Bit though?

Edit:
Actually as I'm doing it as integration by parts I could make that the part you differentiate right?

 

[Ray Vickson]

that makes sense, thank you.

How do I actually integrate the

e^(-x^2/a^2)

Bit though?

Edit:
Actually as I'm doing it as integration by parts I could make that the part you differentiate right?

Change variables to y = x/a, so you have a density f_Y(y) = a^3 A y^2 e^{-y^2}, so you need to deal with the standard integral \int_{-\infty}^{\infty} e^{-y^2}\, dy. You can find this in many reference sources.

RGV

 

[kai_sikorski]

Are you sure this wasn't discussed in your lecture or text?

http://en.wikipedia.org/wiki/Gaussian_integral

 

[chris_avfc]

Change variables to y = x/a, so you have a density f_Y(y) = a^3 A y^2 e^{-y^2}, so you need to deal with the standard integral \int_{-\infty}^{\infty} e^{-y^2}\, dy. You can find this in many reference sources.

RGV

Are you sure this wasn't discussed in your lecture or text?

http://en.wikipedia.org/wiki/Gaussian_integral

I don't remember ever seeing it in the lectures, but there something similar on the bottom of the question sheet.
I've attached it, although I'm not entirely sure how to use it.

I'm guessing its due to the other constants involved?

 

[Ray Vickson]

I don't remember ever seeing it in the lectures, but there something similar on the bottom of the question sheet.
I've attached it, although I'm not entirely sure how to use it.

I'm guessing its due to the other constants involved?

You're joking, right? The formula you need is right there, in plain view in your attached sheet.

RGV

 

[chris_avfc]

You're joking, right? The formula you need is right there, in plain view in your attached sheet.

RGV

Clearly missing something obvious here, but none of them actually match with the one I've been given, which is to be expected as otherwise it would be too simple, but I still don't get it.

 

[kai_sikorski]

How is the 3rd equation not exactly what you need... you don't even need to do integration by parts or u substitution.

 

[chris_avfc]

How is the 3rd equation not exactly what you need... you don't even need to do integration by parts or u substitution.

Oh god, I was reading the a and the alpha as the same thing, maybe I should just give up the degree now haha.
Right that is so much simpler now.

Before I make another stupid mistake let me check something else with you.
I then need to find the average position of the object, would this be

∫ x P(x) dx with the limits of +∞ and -∞?

 

[chris_avfc]

Oh god, I was reading the a and the alpha as the same thing, maybe I should just give up the degree now haha.
Right that is so much simpler now.

Before I make another stupid mistake let me check something else with you.
I then need to find the average position of the object, would this be

∫ x P(x) dx with the limits of +∞ and -∞?

Sorry to be a pain, but could anyone just confirm with me if my question about the average position is right or wrong?

 

[kai_sikorski]

Yes. I highly suggest you read your course text/notes. This is an absolutely fundamental fact. You can not hope to pass a probability course without knowing this; or for that matter learn something you might apply in your discipline whatever it may be...

 

[chris_avfc]

Yes. I highly suggest you read your course text/notes. This is an absolutely fundamental fact. You can not hope to pass a probability course without knowing this; or for that matter learn something you might apply in your discipline whatever it may be...

Yeah I've got a lot of work to do, I need practice more than anything so I can start to notice the techniques to use and that.
Am I right about how to find the average position though?

Thanks for all the help by the way mate.

 

[kai_sikorski]

Yes you are