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Consider a sports team who scores 1 point goals as a Poisson process of rate $\alpha$ and 3 point goals as an independent Poisson process of rate $\beta$. What is the probability that their their first two goals are both 1 point goals? What is the probability that the team scores two 1 point goals before they score two 3 point goals?

Anyone can help me to explain the structure of the question? This topic is new for me.

user786
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1 Answers1

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This question relies on the relationship between the Poisson and Exponential distributions (see this post). Here's one way to approach the first problem:

  1. In order for the first two goals to be 1-point goals, at least two 1-point goals must occur before the first 3-point goal.
  2. If you knew the time of the first 3-point goal, could you calculate the probability of two or more 1-point goals occuring before that?
  3. How can you move from the probability of two 1-point goals occurring before the first 3-point goal, conditional on the time of the first 3-point goal, to the full probability of interest? (Hint: try to integrate using the probability density function for the time of the first 3-point goal).

Hope this helps!

Chai
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  • Thanks for your reply. I understand your view. Can you explain your third point. – user786 Jan 29 '20 at 18:52
  • I was thinking you could use the law of total probability there. You have the probability of the first two goals being 1-point, conditional on the time at which the first 3-point goal occurs. You know the probability density function for the time at which the first 3-point goal occurs (it's exponential). Combine the two, and you should get what you're looking for. – Chai Jan 29 '20 at 19:52