I'm studying probability and the form of a problem that we are working on is:
$x^2 + y^2 < n/k$, the question being what is the probability that the equation is less than (or greater than) the $n/k$, given intervals for $x$ and $y$ such as
$0 < x <1$, $-1 < y < 1$ and $n$ and $k$ are just two random values.
$n$ and $y$ are just chosen values. Distribution is not specified
A concrete example:
What is the probability that $x^2 + (1-y)^2 < 1/9$
I had started using joint density functions and integrating with respect to $x$ and $y$ using the intervals as the support for the integration limits. Doing that gets me $8/9$ but I'm not certain if that is indicative of $8/9$ < $1/9$, so the probability is just $0$ or what??? 🙂 Any help is greatly appreciated.
Thanks!
Best Answer
You are looking at equation of an circle with restricted domain, This can be easily plotted as,
The area of the points in the domain which satisfy the condition $$x^2+(1-y)^2\lt\frac{1}{9}$$ is
$$\frac{1}{4}\pi\frac{1}{3^2}=\frac{\pi}{36}$$
The area of the domain is $2\cdot1=2$.
Assuming uniform distribution, the required probability is the ratio of these areas, that is, $$\frac{\frac{\pi}{36}}{2}=\frac{\pi}{72}$$