91Ó°ÊÓ

Observe that random variables X and N are independent.

We observe that N has geometric distribution with parameter of success 1 - p₀ .

Hence, $P(N=n)={p_0}^{n-1}.(1-p_0)$.

This is because of the fact that N = n, it implies that in first n - 1 trials we have obtained outcomes equal to zero and in the nth trial we have obtained any other trial rather than zero.

Whatever is the time of first non zero outcome, every other non-zero outcome has proportional probability to be that outcome,

$P(X=j)=\frac{p_j}{1-p_0}$

Observe that we divide p_j with 1 - p₀since that we know considered outcome cannot be zero and we want sum of all probabilities is equal to one.

We have that,

$P(X=j,N=n)=P(X=j/N=n).P(N=n)$

Since the probability $P(X=j)$is equal no matter which time of first non zero outcome is, we have that $P(X=j/N=n)=P(X=j)$

Hence, $P(X=j,N=n)=P(X=j).P(N=n)$.

So, we have proved that X and N are independent.

Yes, as its known what is the outcome of first non-zero outcome does not imply when the first non zero outcome might happen. Hence, N is independent of X.

Yes, as its know the time the first non zero outcome does not tell us what is that outcome. The probabilities remain the same. Hence, X is independent of N.