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The standard deviation is a measurement that shows how spread out the values in a data set are. For a Bernoulli random variable, which can only be 0 or 1, the standard deviation gives us an idea of how much the outcomes differ from the expected value. This helps us understand how varied the results are when you repeat the same experiment many times.
In mathematical terms, the standard deviation, often represented by the symbol \( \sigma \), is calculated by taking the square root of the variance. For Bernoulli variables, the variance is \( p(1-p) \), where \( p \) is the probability of the variable being 1.
To find the standard deviation, you compute:

• The variance, \( \text{Var}(X) = p(1-p) \).
• The square root of the variance, \( \sigma = \sqrt{p(1-p)} \).

Understanding the standard deviation helps you predict how much variation to expect in repeated trials, making it a powerful tool in statistics.

Variance is a key concept in probability and statistics that tells us how much the values of a random variable deviate from the average value. For a Bernoulli random variable, the variance is calculated to show how much we can expect the results to vary when the experiment is repeated.
To calculate the variance of a Bernoulli random variable \( X \), you can use the formula:

• \( \text{Var}(X) = \mathbb{E}(X^2) - (\mathbb{E}(X))^2 \).

For Bernoulli random variables, since \( X^2 = X \), we have that \( \mathbb{E}(X^2) = \mathbb{E}(X) = p \).
This leads to the variance being:

• \( \text{Var}(X) = p - p^2 = p(1-p) \).

Having a clear understanding of variance allows us to interpret the reliability and consistency of our probability model.

Probability rules are guidelines that help us calculate the likelihood of different outcomes in random events. These rules are essential for solving problems involving random variables like the Bernoulli model. The probability rules are useful for making predictions and ensuring the calculations are correct.
For Bernoulli random variables:

• Probability of success (outcome 1): \( p \)
• Probability of failure (outcome 0): \( 1-p \)

The total probability is always 1, such as \( p + (1-p) = 1 \). These rules ensure that all possible outcomes of a random variable are accounted for, making it easier to compute expectations, variances, and standard deviations. They form a foundation for more complex statistical models.

The expected value of a random variable is a fundamental concept in probability, providing the average outcome of a random process. For a Bernoulli random variable, the expected value indicates the average number of times we expect to get an outcome of 1 in repeated trials.
The expected value of a Bernoulli random variable \( X \) is given by:

• \( \mathbb{E}(X) = 1 \cdot p + 0 \cdot (1-p) = p \).

This result tells us that the expected value, or mean, is simply the probability \( p \) of the outcome being 1.
Understanding expected value helps in assessing the long-term average behavior of random variables and is crucial for analyzing patterns in statistics and risk management.