The chance to win in Lottery game is $0.1$. Michael decided to buy a ticket every week until he will win or until he will buy 5 tickets.
if X is the number of weeks Michael bought a lottery ticket,
what is the variance of $X$?
So I Calculated for X=5, which means LLLLL or LLLLLW. I calculated the probabilities for both, added them up together and then calculated the variance by the formula $\frac{1-p}{p^2}$. I get $0.798$ which I'm not sure makes sense, am I doing something wrong? Do I need to compute all weeks? Because if so, in the end I get 1.
Best Answer
The issue is that $X$ does not have a geometric distribution, precisely because you have the extra condition that $X \leq 5$. e.g. A geometric distribution would have $\mathbb{P}(X = 6) > 0$, but clearly $X$ can never be 6.
So $X$ is a discrete random variable that takes values in $\{1,2,3,4,5\}$. Let $Y$ be a geometrically distributed random variable with $p = 0.1$. If $k \leq 4$, then $X$ is behaving geometrically, i.e. $$ \mathbb{P}(X = k) = \mathbb{P}(Y = k) = (1-p)^{k-1}p $$ But we have $$ \mathbb{P}(X = 5) = \mathbb{P}(Y \geq 5). $$ Can you see now how to proceed computing statistics for $X$?