# Joint density function of iid exponential distribution?

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Let random variables X1, X2 and X3 be independent and identically distributed according to the exponential distribution with rate λ. Let Y1=X1, Y2=X1+X2, and Y3=X1+X2+X3.

(a) Find the joint density function of Y1, Y2 and Y3.

(b) Find the marginal density of Y3.

What do I do here? I don't really know where to start. Thank you for any help!

(a) Find the joint density function of Y1, Y2 and Y3.

(b) Find the marginal density of Y3.

What do I do here? I don't really know where to start. Thank you for any help!

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Let random variables X1, X2 and X3 be independent and identically distributed according to the exponential distribution with rate λ. Let Y1=X1, Y2=X1+X2, and Y3=X1+X2+X3.

(a) Find the joint density function of Y1, Y2 and Y3.

(b) Find the marginal density of Y3.

What do I do here? I don't really know where to start. Thank you for any help!

**flumefan1**)Let random variables X1, X2 and X3 be independent and identically distributed according to the exponential distribution with rate λ. Let Y1=X1, Y2=X1+X2, and Y3=X1+X2+X3.

(a) Find the joint density function of Y1, Y2 and Y3.

(b) Find the marginal density of Y3.

What do I do here? I don't really know where to start. Thank you for any help!

P(Y1=y1 AND Y2=y2 AND Y3=y3)

Are Y1,Y2,Y3 independent? If so, this has a nice simplification.

Last edited by RDKGames; 4 days ago

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Joint density function here is just

P(Y1=y1 AND Y2=y2 AND Y3=y3)

Are Y1,Y2,Y3 independent? If so, this has a nice simplification.

**RDKGames**)Joint density function here is just

P(Y1=y1 AND Y2=y2 AND Y3=y3)

Are Y1,Y2,Y3 independent? If so, this has a nice simplification.

I don't know I would assume that they are as X1 X2 and X3 are independent but I don't see how Y1 Y2 and Y3 exist? What's the point of them. Why can't we work with the X's?

Last edited by flumefan1; 4 days ago

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How did you get that?

I don't know I would assume that they are as X1 X2 and X3 are independent but I don't see how Y1 Y2 and Y3 exist? What's the point of them. Why can't we work with the X's?

**flumefan1**)How did you get that?

I don't know I would assume that they are as X1 X2 and X3 are independent but I don't see how Y1 Y2 and Y3 exist? What's the point of them. Why can't we work with the X's?

Just check the def you have. From Wikipedia it says the JOINT pdf is the derivative of the CDF with respect to each variable.

So that’s what you need to find first.

F(y1,y2,y3) = P(Y1<y1 , Y2<y2 , Y3<y3)

Last edited by RDKGames; 4 days ago

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Ignore the above I forgot the exponential is a continuous r.v.

Just check the def you have. From Wikipedia it says the JOINT pdf is the derivative of the CDF with respect to each variable.

So that’s what you need to find first.

F(y1,y2,y3) = P(Y1<y1 , Y2<y2 , Y3<y3)

**RDKGames**)Ignore the above I forgot the exponential is a continuous r.v.

Just check the def you have. From Wikipedia it says the JOINT pdf is the derivative of the CDF with respect to each variable.

So that’s what you need to find first.

F(y1,y2,y3) = P(Y1<y1 , Y2<y2 , Y3<y3)

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Ah okay thank you. How do I find those?

**flumefan1**)Ah okay thank you. How do I find those?

Consider a two variable case.

If X1 takes on value x1 then the above is the same as

Exploit the fact that X1,X2 are iid and compute.

Last edited by RDKGames; 4 days ago

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Have you not seen examples?

Consider a two variable case.

If X1 takes on value x1 then the above is the same as

Exploit the fact that X1,X2 are iid and compute.

**RDKGames**)Have you not seen examples?

Consider a two variable case.

If X1 takes on value x1 then the above is the same as

Exploit the fact that X1,X2 are iid and compute.

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I probably have seen examples, but not understood them. Is it a triple integral? if so how do I solve triple integrals?

**flumefan1**)I probably have seen examples, but not understood them. Is it a triple integral? if so how do I solve triple integrals?

Can you compute double integrals?

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Yes it’s a triple integral. If you can do the double integral above then it’s just one extra integration step for the triple integral.

Can you compute double integrals?

**RDKGames**)Yes it’s a triple integral. If you can do the double integral above then it’s just one extra integration step for the triple integral.

Can you compute double integrals?

=1 I can solve that to find k.

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#10

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I think so, I don't really know what they are as I'm in first year and they are a second year concept but I know how to compute them e.g.

=1 I can solve that to find k.

**flumefan1**)I think so, I don't really know what they are as I'm in first year and they are a second year concept but I know how to compute them e.g.

=1 I can solve that to find k.

What have you covered in your course?

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So your course doesn’t cover multiple integrals? Okay, a different approach is probably expected then.

What have you covered in your course?

**RDKGames**)So your course doesn’t cover multiple integrals? Okay, a different approach is probably expected then.

What have you covered in your course?

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It covers multiple integrals. We briefly touched on them last semester but were told to ignore them as they'd come in second year. However they've come up in my first year stats module. I spoke with calculus lecturer and I know how to do them now sorry for the confusion!

**flumefan1**)It covers multiple integrals. We briefly touched on them last semester but were told to ignore them as they'd come in second year. However they've come up in my first year stats module. I spoke with calculus lecturer and I know how to do them now sorry for the confusion!

Anyway, look at this two variable case. Since are iid it means that

So the joint CDF is

Integration in multiple variables works just like PARTIAL differentiation, just backwards of course.

You don't want to integrate with respect to first because it appears in one of the bounds.

Instead, integrate w.r.t and obtain

Then integrate the leftover stuff w.r.t

The joint PDF is then

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