A bullet is fired at a target.
(X,Y) is a bivariate normal random variable that represent the horizontal and vertical distance from the center of the target where the bullet strikes the target when then gun is aimed at the center of the target.
Assume that X and Y are independent random variables, and that the probability of the bullet landing on any point of the xy-plane depends only on the distance of the point from the target $X^2 + Y^2$.
Show that X and Y are bivariate normal r.v.
Ok... Now they give the solution to the problem.
first because the probability depends only on the $X^2+Y^2$ that means that the joint pmf is some function of $X^2+Y^2$, that is:
$$f_{XY}(x,y) = g(x^2 + y^2)$$
Further because X and Y are independent random variables, thus:
$$f_{XY}(x,y) = f_{x}(x) f_{y}(y)$$
combining these two results yields:
$$f_{X}(x) f_{Y}(y) = g(x^2 + y^2)\tag{1}$$
Now we take the derivative of this equation with respect to "x" and we get:
$$f_{X}'(x) f_{Y}(y) = 2~x~g'(x^2 + y^2)\tag{2}$$
Now dividing (2) by (1) yields:
$$\frac{f_{X}'(x)}{2 x f_{X}(x)} = \frac{g'(x^2+y^2)}{g(x^2+y^2)}$$
at this point they claim that the right hand side is a constant:
$$\frac{g'(x^2+y^2)}{g(x^2+y^2)} = c$$
and now i'm lost because I don't understand why the right hand side is a constant? any ideas?
The book's exact explanation is this:
"The left hand side depends only on x. whereas the right-hand side depends only on $x^2 +y^2$, thus, $\frac{f_{X}'(x)}{xf_{X}(x)} = c$, where c is a constant"
why is it a constant?
Hint: Do the same thing with $\ y\ $ that you did with $\ x\ $.