the price of a stock increases with the probability $1/3$, and decreases with probability $3/2$ daily. Let U be the number of days it goes up, after 5 days. What is $var(U)$?
I think the expected value of U is 5 times 1/3 equals to 5/3, but I am not sure if using e.g. $(1 - 5/3) ^ 2 * 1/3 + （2 - 5/3）^ 2 * 1/9 + ...$ is a correct way of finding the variance.
The probability is going up is the same everyday.
Hence the variance is just $5\left( \frac13\right) \left( \frac23\right)$ according to binomial distribution.