The game is that you keep tossing the coin until you toss the tails. Call the total number of tosses the random variable K.
For example:
- HHT -> K=3
- HHHHHT -> K=6
- T -> K=1
so,
- Calculate the PMF for K, Pk(k) = P(K=k)
- Calculate the expected value of K, E[K]
- After each game, you are paid K dolar. What is the expected value of your earnings? How much woul you pay to play one round of this game?
for first question: PMF for K, Pk(k) = P(K=k)
K=1,2,3..... K be the total number of coin tosses.
$P(K=1)=P(T)=p$
$P(K=2)=P(HT)=(1-p)^2p$
$P(K=3)=P(HHT)=(1-p)^3p$
....
$P(K=n)=P(HHHHHH....HT)=(1-p)^{n-1}$
so what can i do?