Proof verification

[sullen juniper]

Given U-Unif(0,1) and U=u and X-Pois(u) compute E[X].

I could think of a way to compute this but it seems fairly convoluted.

$P(X=k|U=u)=u^k e^{-u}/ k!$ and using the law of total probability (? I guess not sure what exactly this is using the pdf of U seems weird but I saw it in https://math.stackexchange.com/a/567234)

$P(X=k)=\int_0^1 P(X=k|U=u)f_U(u) du=\int_0^1 u^k e^{-u} / k! du $

And lastly sum over this but this integral seems ugly to compute out it looks like an incomplete gamma function but the problem I was solving expects a numerical answer and im not sure how to arrive at that through this approach.

The MSE post mentions using function transformation (MGF, CF etc) but again not clear where that would enter the picture here.

Would appreciate any general advice for problems like these

[odd elbow]

1. Do not ping the Moderators, unless someone is breaking the rules.
2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
3. Do not ping other members randomly for help.
4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
5. Wait patiently for a helper to come along.
6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[unborn saffron]

you mean it's compound uniform where the parameter is uniformly disitrubted?

[covert onyx]

Da Big B

[sullen juniper]

Idk what the stats terminology is exactly

But yea I suppose that is what it is

The parameter for the poisson variate is uniformly randomly distributed

[unborn saffron]

this is correct

[sullen juniper]

I imagined so but it does seem like an ugly way to approach this problem

[elfin imp]

what would the sum over this integral be?

Might be possible to simplify that instead of working out the integral first

[sullen juniper]

1 ofc

[unborn saffron]

but you want EX

[sullen juniper]

Ya

[unborn saffron]

$$ EX = \int _0^\infty \mathbb P{X> t}dt $$

[covert onyx]

aL

[sullen juniper]

Any thoughts on using a CF or MGF

[unborn saffron]

you have a double integral

[elfin imp]

this part

[sullen juniper]

Ya k tmes the sum over k ofc

[unborn saffron]

also an option

compound MGF is well known

[sullen juniper]

Oh ya I've seen this for positive support dist

[sullen juniper]

I don't remember lol is it just function composition?

[unborn saffron]

$$ M_X(t) = E\exp (tX) $$

[covert onyx]

aL

[sullen juniper]

Ya I know this I meant the compound case what do you need to do there would it just be formally making a choice u and then just using that as a parameter for X

But then what? I feel like we haven't made use of therandomized parameter

[elfin imp]

you can simplify this

$\sum_{k\geq 0}kx^k/k!$

[covert onyx]

Coffey

[unborn saffron]

it's a derivative of a power series

[sullen juniper]

Ok so shift the sum inside and geometric thing

I'll try this problem again in some time and check it out

[sullen juniper]

Yup that too

[elfin imp]

apply the xD transform

[unborn saffron]

I'll just leave this here, make of it what you will

[sullen juniper]

Ok yea is easier than I thought it was

$$\begin{aligned}
E[X]&=\sum_{k=0}^\infty k P(X=k)\&= \int_0^1 \sum_{k=1}^\infty \frac{u^k}{(k-1)!} e^{-u} du\
&= \int_0^1 \sum_{k=0}^\infty \frac{u^{k+1}}{k!} e^{-u} du\
&= \int_0^1 u \cdot e^{u} \cdot e^{-u} du = \frac{1}{2}
\end{aligned}$$

[unborn saffron]

1 sec, something was wrong with it

[elfin imp]

it's recommended to keep in mind what these do :
`x * derivative
primitive /x

• 1/(1-x)
  1/(1-the function)
• (1-x^n)/(1-x)
  e^the function`

[covert onyx]

Da Big B

[elfin imp]

from k=1?

[sullen juniper]

what is -1!

[unborn saffron]

he shifted index

[sullen juniper]

Thanks for the help idk why i didnt just do the obvious tihng it just looked ugly at first glance

[sullen juniper]

What exactly is this btw? Is it the Law of Total Probability?

[unborn saffron]

law of total expectation

[sullen juniper]

Ive never seen it for a mixture of continuous and discrete probability dist like this

[unborn saffron]

it's a consequence of the law of total probability

[sullen juniper]

Ah ok

So you can mix the measures used?

[unborn saffron]

this process is called "compounding"

when you take a distribution, say poisson and let its parameter not be fixed but some random variable instead

[sullen juniper]

Ok thanks never encountered it before ill read up on it properly once

[unborn saffron]

[sullen juniper]

How do i close lol

-close

[unborn saffron]

+close

[cursive tendon]

You are not authorised to close this help request!

[sullen juniper]

--close

+close

[cursive tendon]

Please thank your Helpers before closing!

Please thank the helpers who assisted you by clicking the buttons below. You can thank each helper only once. Once you're done, click "Close Post" to close this thread.

[cursive tendon]

Thank you for your feedback! aL has been awarded 1 . They now have 845 . They have 3 daily left for today.

[sullen juniper]

845 bombaclat

[unborn saffron]

thank coffey too you lazy mofo :D!

[elfin imp]

close the post

Noone closes it ffs

everyone reads it but doesn't close it

+close