I am preparing for my Probability's exam, after going through Continuous Distributions, I decided to attempt some exercise questions, and heres a question that has left me completely clueless.
It states that,
The arrival time of People entering in an ice parlor are poisson distributed with rate of 50/hour. Any particular person buy an ice cream with probability of 1/4.
What is the distribution of time until the first scoop of ice cream is sold?
What is the probability that no ice cream is sold in a particular hour?
As both the questions are related to Time that means, I believe, Exponential distribution should be used, with parameter 50, but I am not sure how to incorporate the probability of buying an ice cream in my solution!
Kindly guide me, thank you!
Regards,
Selecting events from a Poisson process with rate $r$ with probability $p$ results in a Poisson process with rate $pr$.
Edit in response to the comments: I'm not sure I understand your questions correctly – my impression is that you seem to be sometimes confusing the Poisson distribution of the number of ice creams sold per hour and the exponential distribution of the time until the first ice cream is sold.
The number of ice creams sold in an hour follows a Poisson distribution with parameter $\lambda=pr\cdot1\mathrm h=\frac14\cdot50\mathrm h^{-1}\cdot1\mathrm h=12.5$. Thus the probability of no ice cream being sold in a particular hour is $(12.5)^0\mathrm e^{-12.5}/0!=\mathrm e^{-12.5}\approx3.7\cdot10^{-6}$.
The time until the first ice cream is sold follows an exponential distribution with parameter $\lambda=pr=\frac14\cdot50\mathrm h^{-1}=12.5\mathrm h^{-1}$ and thus has the probability density function $12.5\mathrm h^{-1}\mathrm e^{-12.5\mathrm h^{-1}t}$ and the cumulative distribution function $1-\mathrm e^{-12.5\mathrm h^{-1}t}$.