Convergence of sequences of RVs

Hi all,

I have a problem which is very confusing to me. Can you please give me
any hints?

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Xn is the number of customers an ATM served up to discrete instant of
time n. Xn is a Binomial distribution:

P(Xn=k)= n_C_k * p^k *(1-p)^(n-k).

Asume that at time instance N, the ATM breaks down and therefore the
customer number the ATM served will remain XN thereafter.

N is a random variable with geometric distribution with mean 100.

Does the sequence converge almost surely and if so to what?
-----------------------------------------------

This problem is really confused to me since Ias I understand it, Xn
obviously converges to XN and the probability of this event is equal
to the probability of breaking down of the ATM which is 1.

Thus Xn --> XN with prob = 1 and Xn converges almost surely.

If my thought is reasonable, why do they give some other information
such as Xn is Binomial and N is Geometric. ...??? I am very confused.

Can you please confirm if my solution is correct?

Thanks

.

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