I have a random variable that gives me the number of hydrogen molecules in a spherical region. The random variable has a poisson pmf.
Now I have to define a new Random variable as the distance from the origin to the nearest molecule and find its pdf.
I am unable to think how to do this. I am unable to think of a way to relate the two random variables i.e. distance and number of molecules in the sphere?
Kindly guide me how to approach this problem. Thanks in advace.
Let the number in a ball of radius $R$ have Poisson distribution with parameter $\lambda$.
Let $X$ be the distance from the origin, or any point $P$, to the nearest molecule. Then $X\gt x$ if and only if the ball of radius $x$ with centre $P$ has no molecules. The number of molecules in a ball of radius $x$ has Poisson distribution parameter $\frac{\lambda x^3}{R^3}$. The probability this is $0$ is $\exp(-\lambda x^3/R^3)$.
So the cumulative distribution function of $X$ is $1-\exp(-\lambda x^3/R^3)$ (for $x\gt 0$). For the density function of $X$, we differentiate.