# Worksheet 04: Bernoulli Distribution

Suppose the random variable \(X\) follows a Bernoulli\((p)\) distribution with unknown parameter \(p\) such that \(0 \leq p \leq 1\), \(X \sim \text{Bernoulli}(p)\). Show that the expectation of \(X\) is equal to \(p\), \(\mathbb{E}[X] = p\).

The density function of the Bernoulli distribution is

\[f(x | p) = p^x (1 - p)^{1-x}\]

Think of and describe a random process that has only two outcomes. Suppose we label the outcome of more interest with a \(1\).

What is a reasonable probability, \(p\), of the outcome labeled with a \(1\)?

Draw a plot of the density function for this distribution.

Since distributions generate data, write down 10 possible values that this distribution is likely to generate?

Suppose the random variable \(X\) follows a Bernoulli\((p)\) distribution with unknown parameter \(p\) such that \(0 \leq p \leq 1\), \(X \sim \text{Bernoulli}(p)\). Show that the variance of \(X\) is equal to \(p(1 - p)\), \(\mathbb{V}[X] = p(1 - p)\).

Think of and describe a random process that has only two outcomes, but where one outcome is

**much more likely**than the other. Suppose we label the outcome of more interest with a \(1\).What is a reasonable probability, \(p\), of the outcome labeled with a \(1\)?

Since distributions generate data, write down 10 possible values that this distribution is likely to generate?

Now pretend that \(p = 1/2\). What might 10 new values from this distribution look like?

For which value of \(p\) would the variance be greater? How is this reflected in the data?