In a football game, John scored at the rate of a $1$ goal every 91.3 minutes he was on the pitch. Assume that his goals occurred in a Poisson process.
$a)$ Write down the pdf of Poisson random variable with parameter $\mu.$
$b)$ Calculate the probability that John doesn't score during first $45-minute$ half of a game.
My Working:
$a)$ Let $X$ be random variable representing the number of minutes John needs to make $1$ goal then the pdf of such Poisson random variable is given by $f(x)=\frac{\mu^{x}{e^{-\mu}}}{x!}$. Now I know this is the general form of Poisson distribution but the data in the problem of $1$ goal every $91.3$ minutes is something I can't seem to put in the the general form of above Poisson distribution. Obviously the parameter $\mu$ is the mean and is unknown to us. We are left with $x$ which represents the number of occurrences. kindly guide me how do I approach the problem and what information I am missing out.
$b)$ As for this part I think once I get the $pdf$, all I need will be to find out $1-Pr(X\leq45)$. Is my working Okay? Kindly guide me