iii) Conditioning on and apply total probability
iv) After you're done with iii), think about this: is the type of the second bird that arrives dependent of the first bird that arrives?
v) Ducks and swans are independent of each other, aren't they?
I don't know how to approach part of this question:
A boy throws a piece of bread to each bird that swims by.
There are 2 types of birds: ducks and swans.
Ducks and swans arrive in independent Poisson processes of rates a and b per minute, respectively.
Let N(t) be the number of pieces of bread thrown by the boy from the time he arrives at the river bank, at time 0, until t minutes later. Let Td be the time until the first duck arrives.
i) Name the distribution of N(t) and state it's mean
I got this as N(t) ~ Poi (at + bt), where the mean is (at + bt)
ii) Name the distribution of Td and state it's mean
I got Td ~ exp (a), where the mean is 1/a
iii) Find the probability that the first bird to arrive is a duck
This is where I get stuck. I figure that this is P(Td < Ts), but then how would I do this?
And then there's a couple more parts after it which I can't figure out:
Consider the first 20 pieces of bread thrown by the boy. Let Nd be the number of these 20 pieces that are eaten by a duck.
iv) Find the distribution of Nd and explain why it is.
v) Find the probability that after 10 minutes, the boy has thrown a piece of bread to at least one duck and at least one swan.