11.3 Mean and Variance of the Bernoulli Distribution

The Bernoulli distribution models a random experiment with exactly two possible outcomes: success (typically represented as 1) and failure (typically represented as 0). It's a fundamental concept in probability theory, forming the basis for more complex distributions like the Binomial distribution. Understanding its mean and variance is crucial for analyzing binary events.

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I. Mean (Expected Value) of the Bernoulli Distribution

The mean, denoted by $\mu$ or $E[X]$, represents the expected value or average outcome of a Bernoulli random variable over a large number of trials. For a Bernoulli distribution, the mean is simply the probability of success.

Formula

If $X$ is a Bernoulli random variable with:

• $P(X = 1) = p$ (probability of success)
• $P(X = 0) = q = 1 - p$ (probability of failure)

The mean is calculated as:

$$ \mu = E[X] = p $$

Explanation

The expected value is calculated by summing the product of each possible outcome and its corresponding probability:

$$ E[X] = (1 \times P(X = 1)) + (0 \times P(X = 0)) $$ $$ E[X] = (1 \times p) + (0 \times q) $$ $$ E[X] = p + 0 $$ $$ E[X] = p $$

This intuitively means that, on average, the outcome will be closer to the probability of success. If an event has a 70% chance of success ($p=0.7$), then over many trials, you would expect the average outcome to be 0.7.

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II. Variance of the Bernoulli Distribution

The variance, denoted by $\sigma^2$ or $Var[X]$, measures the spread or dispersion of the possible outcomes from the mean. For a Bernoulli distribution, the variance quantifies how much the results tend to deviate from the expected value ($p$).

Formula

The variance of a Bernoulli distribution is given by:

$$ \sigma^2 = Var[X] = p \times (1 - p) $$ $$ \sigma^2 = pq $$

Derivation

The variance can be derived using the formula: $Var[X] = E[X^2] - (E[X])^2$.

First, let's calculate $E[X^2]$:

$$ E[X^2] = (1^2 \times P(X = 1)) + (0^2 \times P(X = 0)) $$ $$ E[X^2] = (1 \times p) + (0 \times q) $$ $$ E[X^2] = p $$

Now, substitute this and $(E[X])^2$ into the variance formula:

$$ Var[X] = E[X^2] - (E[X])^2 $$ $$ Var[X] = p - (p)^2 $$ $$ Var[X] = p - p^2 $$ $$ Var[X] = p(1 - p) $$ $$ Var[X] = pq $$

Interpretation of Variance

The variance $p(1-p)$ highlights that the spread of outcomes is dependent on both the probability of success ($p$) and the probability of failure ($q$).

• Maximum Variance: The variance is maximized when $p = 0.5$. In this case, $Var[X] = 0.5 \times (1 - 0.5) = 0.25$. This occurs when success and failure are equally likely, leading to the greatest uncertainty in outcomes.
• Minimum Variance: As $p$ approaches 0 or 1 (meaning the outcome is almost certain to be failure or success, respectively), the variance approaches 0. For example, if $p=0.99$, $Var[X] = 0.99 \times 0.01 = 0.0099$. If $p=0.01$, $Var[X] = 0.01 \times 0.99 = 0.0099$. This indicates very little variability when one outcome is highly probable.

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Summary of Properties

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This understanding of the mean and variance of the Bernoulli distribution is foundational for many statistical analyses, data science applications, and business decision-making processes involving binary outcomes.

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Frequently Asked Questions

• What are the key properties of the Bernoulli distribution? The key properties are that it has only two possible outcomes (success/failure), it's a discrete probability distribution, and its mean is $p$ and its variance is $p(1-p)$.

• Why is the Bernoulli distribution used for binary outcomes? It's specifically designed to model a single trial of an experiment where there are only two mutually exclusive and exhaustive outcomes, such as a coin flip, a customer making a purchase, or a machine producing a defect.

• How is the probability of failure calculated in a Bernoulli distribution? The probability of failure ($q$) is simply the complement of the probability of success ($p$). Therefore, $q = 1 - p$.

• How do you calculate the expected value of a Bernoulli distribution? The expected value is equal to the probability of success, $p$.

• Can you explain the variance formula of a Bernoulli distribution? The variance is $p(1-p)$, meaning it measures the spread of outcomes. It's highest when $p=0.5$ and approaches zero as $p$ gets closer to 0 or 1.

• How does the Bernoulli distribution relate to the Binomial distribution? The Binomial distribution is essentially a sum of $n$ independent Bernoulli trials. If you have $n$ identical and independent Bernoulli experiments, the number of successes follows a Binomial distribution.