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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.

What is the definition of the Bernoulli distribution and how is it applied to an unbalanced coin toss?

Understand the Bernoulli distribution with parameter p and its application to an unbalanced coin toss. This example includes key formulas for probability, expected value, and variance.

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