I'm bad at stochastics so really glad for any help(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

I have two normally distributed NON INDEPENDENT stochastic variables X~N(muX,sigX^2) and Y~N(muY,sigY^2)

A third variable D is defined as D = sqrt(X^2 + Y^2).

Since Y and X are stochastic D will also be stochastic.

2. Relevant equations

But how to calculate expected value muD and variance sigD^2 properly?

3. The attempt at a solution

Calculate the variance sigD^2

D = sqrt(X^2 + Y^2) (1)

D^2 = X^2 + Y^2 (2)

E[D^2] = E[X^2 + Y^2] = E[X^2] + E[Y^2] (3)

sigD^2 = E[(D - muD)^2] = E[D^2] - muD^2 (4)

Using (3) in (4)

sigD^2 = E[X^2] + E[Y^2] - muD^2 (5)

sigD^2 = E[X]^2 + Var[X] + E[Y]^2 + Var[Y] - muD^2 (5)

sigD^2 = muX^2 + sigX^2 + muY^2 +sigY^2 - muD^2 (6)

So far so good.

Calculate muD

At this point I thought I'm done.

But what about muD? I dont have it!

At first tempting to assume

muD = sqrt(muX^2+muY^2)

But I dont think it's true, since X and Y are not independent. And even if it is true, how to show it.

If I start out

muD = E[D] = E[sqrt(X^2 + Y^2)]

I dont manage to come to a solution.

Really appreciate any help

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# Calculate expected value and variance of d, d = sqrt(x^2+y^2)

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