Worksheet 04: Bernoulli Distribution

Due Date

2023-03-10 Friday at the end of class

1. 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

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

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

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

   2. Draw a plot of the density function for this distribution.

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

3. 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 \(p(1 - p)\), \(\mathbb{V}[X] = p(1 - p)\).

4. 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 \(1\).

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

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

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

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