Hello I am trying to calculate this question,
Every year there is 5 shooting stars.
I am calculating the stars shooting past in the sky and in the first 6 months i have calculated the probability that there will be at least 2 shooting stars.
=2.5 since 5/2 (half a year)
(≥2)=1−(<2)
Now i want to calculate for when there has been at least 4 shooting starts in the first 9 months assuming that the event above (the shooting star in the first 6 month) had occurred
How should i go about this ...
Let $X_{[a,b]}$ denote the number of shooting stars in time interval $[a,b]$ (in months).
It has Poisson distribution with parameter $\frac{5(b-a)}{12}$ and if $[a,b]$ and $[c,d]$ are disjoint then $X_{[a,b]}$ and $X_{[c,d]}$ are independent.
First calculate:$$P(X_{[0,9]}\geq4,X_{[0,6]}\geq2)=$$$$P(X_{[0,9]}\geq 4,X_{[0,6]}=2)+P(X_{[0,9]}\geq4,X_{[0,6]}=3)+P(X_6\geq4)=$$$$P(X_{[0,9]}\geq4\mid X_{[0,6]}=2)P(X_{[0,6]}=2)+P(X_{[0,9]}\geq4\mid X_{[0,6]}=3)P(X_{[0,6]}=3)+P(X_{[0,6]}\geq4)=$$$$P(X_{(6,9]}\geq2)P(X_{[0,6]}=2)+P(X_{(6,9]}\geq1)P(X_{[0,6]}=3)+P(X_{[0,6]}\geq4)$$
In the last equality it is used that $X_{[0,9]}=X_{[0,6]}+X_{[(6,9]}$ and that $X_{[0,6]}$ and $X_{[(6,9]}$ are independent.
Then calculate:$$P(X_{[0,9]}\geq4\mid X_{[0,6]}\geq2)=\frac{P(X_{[0,9]}\geq4,X_{[0,6]}\geq2)}{P(X_{[0,6]}\geq2)}$$