# Stats

• Mar 6th 2008, 08:04 AM
heathrowjohnny
Stats
Let $\displaystyle X_1 \sim N(0,2)$ and $\displaystyle X_2 \sim N(0,2)$ be normally distributed random variables that denote a point in $\displaystyle \bold{R}^{2}$. Suppose that the distance squared from this point to the origin is $\displaystyle D^2 = X_{1}^{2} + X_{2}^{2}$. Find the probability that $\displaystyle D > 3$.

So this is the same as $\displaystyle 1 - P(D^{2} < 9)$. And is $\displaystyle D^{2}$ a $\displaystyle \chi_{2}^{2}$ distribution with $\displaystyle 2$ degrees of freedom?
• Mar 6th 2008, 08:16 AM
colby2152
Quote:

Originally Posted by heathrowjohnny
Let $\displaystyle X_1 \sim N(0,2)$ and $\displaystyle X_2 \sim N(0,2)$ be normally distributed random variables that denote a point in $\displaystyle \bold{R}^{2}$. Suppose that the distance squared from this point to the origin is $\displaystyle D^2 = X_{1}^{2} + X_{2}^{2}$. Find the probability that $\displaystyle D > 3$.

So this is the same as $\displaystyle 1 - P(D^{2} < 9)$. And is $\displaystyle D^{2}$ a $\displaystyle \chi_{2}^{2}$ distribution with $\displaystyle 2$ degrees of freedom?

I believe that is correct, although I am hesitant to verify the degrees of freedom. My statistics knowledge is limited to where I practice it, so let someone like CaptainBlack respond to your question.
• Mar 7th 2008, 02:15 AM
heathrowjohnny
but dont they have to be standard normal variables?
• Mar 7th 2008, 08:18 PM
mr fantastic
Quote:

Originally Posted by heathrowjohnny
but dont they have to be standard normal variables?

Yes.

Quote:

Originally Posted by heathrowjohnny
Let $\displaystyle X_1 \sim N(0,2)$ and $\displaystyle X_2 \sim N(0,2)$ be normally distributed random variables that denote a point in $\displaystyle \bold{R}^{2}$. Suppose that the distance squared from this point to the origin is $\displaystyle D^2 = X_{1}^{2} + X_{2}^{2}$. Find the probability that $\displaystyle D > 3$.
[snip]

Note that $\displaystyle \frac{X_1}{2}$ and $\displaystyle \frac{X_2}{2}$ are standard normal variables. So

$\displaystyle Y^2 = \frac{X_1^2}{2^2} + \frac{X_2^2}{2^2} = \frac{1}{4} \left( X_1^2 + X_2^2 \right)$

will follow a chi-square distribution with 2 degrees of freedom.

You want $\displaystyle \Pr\left( Y^2 > \frac{9}{4} \right)$.

I get 0.32465 as the answer (correct to 5 decimal places).
• Mar 8th 2008, 04:45 PM
mr fantastic
Quote:

Originally Posted by mr fantastic
Yes.

Note that $\displaystyle \frac{X_1}{2}$ and $\displaystyle \frac{X_2}{2}$ are standard normal variables. So

$\displaystyle Y^2 = \frac{X_1^2}{2^2} + \frac{X_2^2}{2^2} = \frac{1}{4} \left( X_1^2 + X_2^2 \right)$

will follow a chi-square distribution with 2 degrees of freedom.

You want $\displaystyle \Pr\left( Y^2 > \frac{9}{4} \right)$.

I get 0.32465 as the answer (correct to 5 decimal places).

I suppose I should add that the exact answer is $\displaystyle e^{-9/8}$, got by integrating the chi-square pdf.
• Mar 23rd 2008, 04:51 PM
mr fantastic
The derivation in this thread might be of background interest here.