Ask Your Question

# Symbolic expectations and double integrals

I'd like to compute the following expectation ($U$ and $V$ are independent and normally distributed or Gaussian) $a_{k,p}=E(|U|^p|U+\sqrt{k-1}V|^p)$

Is there a way to directly compute those expectations in Sage ?

If I write it down, I obtain a double integral which I already tried to compute with maxima like this (ok here, you can only see the starting point with the inner integral with respect to $v$ first)

load(distrib);
n(x):=pdf_normal(x, 0, 1);
inner_integral: integrate(abs(u+sqrt(k-1)*v)^p*n(v), v, minf, inf);


but to no avail.

NB: The absolute central moment $a_{1,p/2}$ can be easily obtained with maxima with

ratsimp(integrate(abs(x)^p*n(x), x, minf, inf));


but no 'simple' expression as for the aforementioned double integral.

Any hint?

edit retag close merge delete

## Comments

Since Sage uses Maxima for all of its symbolic integration, I find it unlikely we'll do better. We do have numerical routines of various types, but I don't think this is what you are looking for.

@kcrisman I thought that sometimes Sage could use sympy or some other CAS for symbolic integration. But then, I'm stuck here then.

## 1 Answer

Sort by » oldest newest most voted

Suppose n(x) was defined somehow (in fact, in extenso) but see also my related question about how to import Maxima built-in/package functions. My current result so far:

absm(p)=integral(abs(x)^p*n(x), x, -infinity, infinity)
absm
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
a(k,p)=integral(integral(abs(u)^p*abs(u+sqrt(k-1)*v)^p*n(v)*n(u), v, -infinity, infinity), u, -infinity, infinity)
a(k,p)
1/2*integrate(abs(u)^p*integrate(abs(sqrt(k - 1)*v + u)^p*e^(-1/2*u^2 - 1/2*v^2), v, -Infinity, +Infinity), u, -Infinity, +Infinity)/pi
ab(p)=a(1,p/2).simplify()
ab
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
bool(absm == ab)
True


This is a partial result but a(k,p) can't be simplified in terms of gamma's or whatnots.

How do you make changes of variables in Sage?

more

## Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

## Stats

Asked: 2012-01-21 09:07:56 +0200

Seen: 825 times

Last updated: Jan 28 '12
{"ft_lang_label":"__label__en","ft_lang_prob":0.83509094,"math_prob":0.9524609,"size":1121,"snap":"2021-21-2021-25","text_gpt3_token_len":290,"char_repetition_ratio":0.10832588,"word_repetition_ratio":0.0,"special_character_ratio":0.2515611,"punctuation_ratio":0.12765957,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9982461,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-25T10:43:50Z\",\"WARC-Record-ID\":\"<urn:uuid:a35a1ae8-b205-4c26-a9a0-d3060be9f119>\",\"Content-Length\":\"58429\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7aea707e-e615-47d3-9c99-5548376e1eda>\",\"WARC-Concurrent-To\":\"<urn:uuid:fcceddd2-6150-4fb6-987a-c8c3b5d1df73>\",\"WARC-IP-Address\":\"194.254.163.53\",\"WARC-Target-URI\":\"https://ask.sagemath.org/question/8648/symbolic-expectations-and-double-integrals/\",\"WARC-Payload-Digest\":\"sha1:WWK22LI2EMY7EE27GUPGRHBD767PVL5C\",\"WARC-Block-Digest\":\"sha1:SDZJXYPP7GHRLY547S7FPOFUT4775PJR\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487630081.36_warc_CC-MAIN-20210625085140-20210625115140-00623.warc.gz\"}"}