Let $X_1 , X_2, X_3, X_4$ be mutually independent and identically distributed Bernoulli(p) random variables. Find $V[X_1 X_2 ^2 X_3 ^3 X_4 ^4]$ and find $P(X_1 \leq X_2 \leq X_3 \leq X_4)$.
A few notes, I know variance for a simple single random binomial variable is np(1-p), yet I struggle to apply that here. Moreover, while I know that the expected value of the product of INDEPENDENT random variables equals the product of the expected values of the random variables, I'm wondering if that applies here to the variances as well?
As for the simple(r) probability portion, given that it has no bounds included in the parameters of the problem, could it simply be taking the triple integral of 1 with the bounds for each integral from negative infinity to infinity? As that would be a very unsatisfying answer given it really provides no information.
Alternatively, for the probability portion, would it not involve some trickery using the standard np(1-p) and exponents?