- #1

Eclair_de_XII

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- 91

## Homework Statement

"A helicopter drops rocks onto a desert until it hits a marked bullseye. With respect to Cartesian coordinates whose origins is at the bullseye, both the x- and y-coordinates are normally distributed. Denote the r.v.'s taking on values along these axes as ##X,Y##, respectively, and note that they are independent. Also note that ##X,Y## have distributions ##N(0,\sigma^2)##. Show that the expectation of the distance between the landing point and the target is ##\sigma \sqrt{\frac {\pi}{2}}##. Find also the variance."

## Homework Equations

Hints given: "Find ##P(X^2+Y^2 \leq r^2)## and use that distribution to find ##P(\sqrt{X^2+Y^2}\leq r)##, ##Var(\sqrt{X^2+Y^2})## and ##E(\sqrt{X^2+Y^2})##.

If ##X## has distribution ##N(\mu,\sigma^2)##, then ##f_X(x)=\frac {1}{\sqrt{2 \pi} \sigma} e^{-\frac {1}{2}(\frac {x-\mu}{\sigma})^2}##.

For two independent r.v.'s ##X,Y##, ##f_{X,Y}(x,y)=f_X(x)f_Y(y)##.

## The Attempt at a Solution

So far, what I have is:

##P(X^2+Y^2 \leq r^2)=2P(Y\leq \sqrt{r^2-x^2}|X=x)P(X=x)=2\int_{-r}^{r} \int_0^{\sqrt{r^2-x^2}} f_Y(y)f_X(x)\,dy\,dx\\

=4\int_0^r \int_0^{\sqrt{r^2-x^2}} \frac{1}{2\pi \sigma^2} e^{-\frac{1}{2\sigma^2}y^2}e^{-\frac{1}{2\sigma^2}x^2}\,dy \,dx=\frac{2}{\pi \sigma^2} \int_0^r \int_0^{\sqrt{r^2-x^2}}e^{-\frac{1}{2\sigma^2}(y^2+x^2)}\,dy \,dx##

This integral isn't something I learned how to do yet, so I'm thinking that there's another way I'm supposed to do this problem.