# Probability theory question: bivariate normal and conditional probability

**URL:** <https://boards.straightdope.com/t/probability-theory-question-bivariate-normal-and-conditional-probability/334984>\
**Category:** Factual Questions\
**Created:** [December 10, 2005, 1:48am UTC](https://boards.straightdope.com/t/probability-theory-question-bivariate-normal-and-conditional-probability/334984 "2005-12-10T01:48:26Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![quelquechose](https://avatars.discourse-cdn.com/v4/letter/q/6f9a4e/32.png) [@quelquechose](https://boards.straightdope.com/u/quelquechose)\
**Post date:** [December 10, 2005, 1:48am UTC](https://boards.straightdope.com/t/probability-theory-question-bivariate-normal-and-conditional-probability/334984/1 "2005-12-10T01:48:26Z")

</div>

First of all, this is from a homework assignment, but I didn’t see any rule forbidding such questions so I figured it was okay to ask.

Suppose a random variable X has a normal distribution, and for every x the conditional distribution of another random variable Y given that X=x is normal with mean a\*x+b and variance t^2 where a,b, and t^2 are constants. Prove that the joint distribution of X and Y is a bivariate normal.

I have tried several approaches to this problem, but so far I haven’t been successful. My first reaction is to multiply the conditional pdf of Y by the marginal on X to get the joint pdf, and then show that this is equivalent to the joint pdf of a bivariate normal. However, for some reason this doesn’t work. Why is this? (algebra below)

Let f[sub]1/sub be the marginal on x, and g(y|x) be the conditional pdf for Y given that X=x. Then the joint pdf f(x,y) is:

f(x,y)=f[sub]1/sub\*g(y|x)

(assume X has mean m and variance s[sub]1[/sub]^2) Substitution gives:

f(x,y)=1/(sqrt(2pi)\*s[sub]1[/sub])_e^(-0.5_((x-m[sub]1[/sub])/s[sub]1[/sub])^2)\*1/(sqrt(2pi)\*t)_e^(-0.5((y-a_x-b)/t)^2)

Since the above expression does not contain rho, I’m not sure how to get it into the standard form for a bivariate normal, which does include rho. I’m not looking for a step by step explanation, but some general suggestions would be helpful.

---

<div class="post-metadata">

**Author:** ![samclem](https://avatars.discourse-cdn.com/v4/letter/s/a9a28c/32.png) [@samclem](https://boards.straightdope.com/u/samclem)\
**Post date:** [December 10, 2005, 2:03am UTC](https://boards.straightdope.com/t/probability-theory-question-bivariate-normal-and-conditional-probability/334984/2 "2005-12-10T02:03:02Z")

</div>

The basic rule around these parts is:

If you want us to DO your homework, that ain’t allowed.

But you are looking for suggestions. That I’ll allow.

**samclem** GQ moderator

---

<div class="post-metadata">

**Author:** ![Becky](https://avatars.discourse-cdn.com/v4/letter/b/c4cdca/32.png) [@Becky](https://boards.straightdope.com/u/Becky)\
**Post date:** [December 10, 2005, 2:17am UTC](https://boards.straightdope.com/t/probability-theory-question-bivariate-normal-and-conditional-probability/334984/3 "2005-12-10T02:17:36Z")

</div>

Hint: rho describes the strength of the linear relationship between X and Y. What quantity that you mentioned above has a similar purpose?

---

<div class="post-metadata">

**Author:** ![quelquechose](https://avatars.discourse-cdn.com/v4/letter/q/6f9a4e/32.png) [@quelquechose](https://boards.straightdope.com/u/quelquechose)\
**Post date:** [December 10, 2005, 2:49pm UTC](https://boards.straightdope.com/t/probability-theory-question-bivariate-normal-and-conditional-probability/334984/4 "2005-12-10T14:49:58Z")

</div>

Rho depends mostly on t. If t is 0 and a is not 0, then rho must be 1. As t gets large rho gets small, but I’m not sure how to use this to determine exactly what rho is in terms of the other variables.