Question

Let’s fix some n e {1,2,3,...}. A biased coin with probability of head p is tossed n times, and the number of heads, N1, is counted. The coin is then tossed Ni more times, and the number of heads, N2, is counted. Find the expected total number of heads, E[Ni + N2], generated by this process. The answer should depend on n and p only.

Answer 1

The answer is "np(1+p)".

Explanation:

`N_1` = Number of n tossed heads

According to
`N_1`, Y = amount of
`N_1` tossing heads

So,
`N_1`~ Bin(n, p) and
`N_2` |
`N_1` ~ Bin(
`N_1`, p)

`E(N_1+N_2) & = E\Big[N_1+N_2\vert N_1\Big]\\ &`

`= E(N_1)+E\Big[N_2\vert N_1\Big]\\\\ & = E(N_1) + E(N_1\,p)\\\\ & = np + np.p \\\\ =np+np^2\\\\ =np(1+p)`