Let Ti be the number of trials up to and including the ith success.
Let Nn be the number of successes in the first n trials. show that
P(T1=x|Nn=1)=1/n x=1,2,...n
thanks.
Let Ti be the number of trials up to and including the ith success.
Let Nn be the number of successes in the first n trials. show that
P(T1=x|Nn=1)=1/n x=1,2,...n
thanks.
$\displaystyle T_1$ ~ Geometric(p).
$\displaystyle N_n$ ~ Binomial(n, p).
Therefore $\displaystyle \Pr(N_n = 1) = n p (1 - p)^{n-1}$ and $\displaystyle \Pr(T_1 = x \cap N_n = 1) = (1 - p)^{x-1} p \cdot (1 - p)^{n-x}$.