I am given the scenario where a biased coin, with probability p € (0,1) of landing on heads is tossed repeatedly. For each n=1,2,..... let Xn be the total number of heads seen after the nth toss.
What is the distribution of Xn?
Is Xn a geometric distribution?
Thanks
Thanks, i understand why it isnt geometric now
The last part of the question says now suppose for k=1,2,.. ...Ak is the event that the first head is seen on the kth toss. Fix n>=1. Show that for all k €(1,2,.....n)
P(Ak| Xn=1)=1/n
Would I start of by writing P(Ak| Xn=1) = P(Ak ; Xn=1) / P(Xn=1)
=P(Ak n Xn=1)/ P(Xn=1)
to calculate P(Xn=1) would I plug this into the binomial formula above, then I'm not sure what to do abou P(Ak n Xn=1)?
thanks for your help
that intersection means that you had only one head in n tosses (Xn=1)
AND that head appeared in the k-th toss (Ak)
SO looking at the n independent tosses, you
had tail, tail, tail..., head on k-th, tail,...tail.
that event has probability q n-1 times and p once.
The ratio is