# Variance of the gain, given a set of wagers on $n$ Bernoulli trials

Let's assume we want to wager on a given set of n Bernoulli trials with probabilities $p_i$ with $i \in \{1, 2, ..., n\}$ and a corresponding payoff odds $b_i$, i.e., betting a certain percentage $f_i \in (0,1)$ of our (entire) bankroll results in either, a win with a wealth increase of $f_i b_i$, or loss with a wealth decrease of $-f_i$.

Our initial capital is $X_0$. Suppose we choose the goal of maximising the expected value $E(X_n)$ after $n$ trials.

Note that $E(X)$ is the expected value for individual trials with $E(X)=(bf) p + (-f)q$.

Now my question is as follows, what is the variance of the gain after $n$ trials? Since we repeatedly invest a percentage $f$ of our bankroll, I figured the expected mean $E(X_n)$, would grow geometrically and thus,

$E(X_n)=E(X_{n-1})E(X_{n-2})...E(X_1)X_0$.

But I am not entirely sure what $Var(X_n)$ then would be.

The reason why I am asking is that I would like to differentiate between normal volatility around the expected mean and deviant behaviour away from it, i.e., losing more of the bankroll than one would suspect.

EDIT By stating the question I realised the answer is given by this post.

• Notational/usage nitpick: $E(X_n)$ is a constant, so $Var(E(X_n))$ isn't what you're looking for. You want $Var(X_n)$. Similarly you want the "variance of the gain after $n$ trials" (no "expected"). I've edited accordingly. – Michael Lugo Jan 25 '17 at 14:14