# Bernoulli Distribution ### Variance Note that the [Variance](Variance.md) of a bernoulli rv is maximized when the parameter (probability of success) is 0.5. This is clear because a bernoulli rv can only have an outcome of 0 or 1. When p is 0.5 it is far from them both, and both 0 and 1 are equally likely to occur. If p is small or large (i.e. as it moves away from 0.5) the value that would increase the variance will be come less likely, thus decreasing the variance overall.  As a simple example consider if p = 0.1 Our expectation would then look like: $ Var(X) = \mathbb{E} \big[ (X - \mu)^2 \big] = \sum_{i \in \{ 0, 1\}} p_i(x_i - \mu)^2 = \color{red}0.9\color{black}(0 - 0.1)^2 + \color{red}0.1\color{black}(1 - 0.1)^2 $ We see that the probabilities (in red) of both 0 and 1 of occurring are directly tied to the underlying $\mu$. This constraint means that when the deviation would be high (say, $1 - 0.1$), the likelihood of observing a $1$ is very low ($0.1$). This pulls this squared deviation down. --- Date: 20220309 Links to: [Probability MOC](Probability%20MOC.md) Tags: References: * []()