Problem 5 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Determine the expectation and variance of the \(\operatorname{Ber}(p)\) distribution.

Step-by-Step Solution

Answer

The expectation is \( p \) and the variance is \( p(1-p) \).

Step 1: Understanding Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution of a random variable which can take the value 1 with probability \( p \) and the value 0 with probability \( 1-p \). This type of distribution models the outcome of a single experiment that can result in one of two possible outcomes: success (1) or failure (0). Hence, only two parameters are involved: the probability \( p \) of success, and consequently \( 1-p \) is the probability of failure.

Step 2: Write Down Expectation Formula

The expectation (mean) of a random variable \( X \) with a Bernoulli distribution \( \operatorname{Ber}(p) \) can be calculated using the formula: \( E(X) = \sum_{x} x \, P(X=x) \). Since the variable can only be 0 or 1, we focus on these two values.

Step 3: Calculate Expectation

Substitute the values 0 and 1 into the expectation formula: \[E(X) = 0 \, \cdot \, P(X=0) + 1 \, \cdot \, P(X=1) = 0 \, \cdot \, (1-p) + 1 \, \cdot \, p = p\]Thus, the expectation \( E(X) \) for a \( \operatorname{Ber}(p) \) distribution is \( p \).

Step 4: Write Down Variance Formula

The variance of a random variable \( X \) is given by the formula: \( \text{Var}(X) = E(X^2) - (E(X))^2 \). Since, for a Bernoulli distribution, \( E(X^2) = p \) because the possible values 0 and 1 satisfy \( X^2 = X \).

Step 5: Calculate Variance

First calculate \( E(X^2) \):\[E(X^2) = 0^2 \, \cdot \, (1-p) + 1^2 \, \cdot \, p = p\]Then find the variance:\[\text{Var}(X) = E(X^2) - (E(X))^2 = p - p^2 = p(1-p)\]Thus, the variance \( \text{Var}(X) \) for a \( \operatorname{Ber}(p) \) distribution is \( p(1-p) \).

Key Concepts

Expectation of a Bernoulli Distribution

The expectation, or mean, is a fundamental concept used to describe the average or central value of a random variable. For the Bernoulli distribution—denoted as \( \operatorname{Ber}(p) \)—this distribution is quite straightforward to understand. The Bernoulli distribution deals with a random event that can result in either a success or a failure. When we say success, it's simply our particular condition of interest turning out to be true, taking a value of \(1\). Any other outcome, not meeting the success condition, is a failure, taking a value of \(0\).

To compute the expectation of a Bernoulli-distributed random variable \(X\), we use the formula:
\[ E(X) = 0 \cdot (1-p) + 1 \cdot p = p \]
The expectation, \( E(X) \), is simply \( p \). It represents the probability of success across many trials. Think of it as the expected "average" value of results from a very large number of these binary trials.

Thus, if you repeat an experiment many times, with each trial being independent and having probability \( p \) of success, you'd expect, on average, successes to occur \( p \) proportion of the time.

Variance of a Bernoulli Distribution

Variance is another essential concept that describes how much the values of a random variable deviate from the expectation (mean). For a Bernoulli distribution, variance helps us understand the variability or spread of the outcomes.

In the context of \( \operatorname{Ber}(p) \), the variance is derived using the formula:
\[ \text{Var}(X) = E(X^2) - (E(X))^2 \]
The calculation of \( E(X^2) \) is simple as it equals \( p \) because the possible outcomes are 0 and 1, and \( X^2 = X \). Therefore, the variance for a Bernoulli random variable \( X \) simplifies to:
\[ \text{Var}(X) = p - p^2 = p(1-p) \]
This result shows us how spread out or variable the outcomes are.

• If \(p\) is close to 0 or 1, the variance is low since outcomes are more predictable.
• If \(p\) is close to 0.5, there's a higher variance, representing more unpredictability in outcomes.

Understanding variance is crucial in statistics because it gives insight into the reliability and consistency of repeated trials of an experiment.

Probability Distribution

A probability distribution is a statistical function that describes all the possible values and their probabilities for a particular experiment. For a Bernoulli distribution, the focus is especially on simple experiments with two possible outcomes.

In the Bernoulli distribution \( \operatorname{Ber}(p) \), the probabilities are:

• The probability of success (outcome 1) is \( p \).
• The probability of failure (outcome 0) is \( 1-p \).

Each trial in this context is independent of the others and results in binary outcomes, making it a perfect model for many real-world scenarios, such as flipping a coin or a single yes/no survey question.

Additionally, knowing how to graphically represent a Bernoulli probability distribution on a bar chart can be insightful. The height of the bars represents probabilities: one bar at \( x = 0 \) with height \( 1-p \), and another at \( x = 1 \) with height \( p \).

Probability distributions, like the Bernoulli, are fundamental to understanding the likelihood of different outcomes in random scenarios, allowing us to make informed predictions and decisions based on statistical data.