# 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?