This image provides a definition of the Bernoulli distribution and an example of its application. Here's a breakdown:

Definition:

The random variable (r.v.) $X$ follows the Bernoulli distribution with parameter $p \in [0,1]$ if it takes only two values, 0 and 1. The probabilities are:

• $\mathbb{P}(X = 1) = p$
• $\mathbb{P}(X = 0) = 1 - p = q$, where $p + q = 1$.

We denote this as $X \sim \mathcal{B}(p)$.

The expected value and variance are:

• $\mathbb{E}(X) = p$
• $\text{Var}(X) = p(1 - p) = pq$

Example:

Consider the toss of an unbalanced coin, where the probability of getting "heads" is $p \neq \frac{1}{2}$. Let $X$ be the random variable such that:

• $X = 1$ if we get "heads"
• $X = 0$ if we get "tails"

In this case, $X \sim \mathcal{B}(p)$, and the probability distribution is:

• $\mathbb{P}(X = 1) = p$ (for heads)
• $\mathbb{P}(X = 0) = q$ (for tails, where $q = 1 - p$).

Would you like more details or explanations on this topic?

Here are 5 relative questions to expand on this:

1. How is the Bernoulli distribution related to the Binomial distribution?
2. Can you explain how to calculate the expected value for a general discrete random variable?
3. What real-life examples can be modeled using the Bernoulli distribution?
4. How does the variance in a Bernoulli distribution behave as $p$ approaches 0 or 1?
5. What is the importance of the Bernoulli distribution in hypothesis testing?

Tip: The Bernoulli distribution is the building block for the Binomial distribution, where multiple independent Bernoulli trials are combined.