Bernoulli Distribution Study Guide for University Statistics Exams

📚 Quick Study Guide

• 🎲 Definition: The Bernoulli distribution models the probability of success or failure of a single trial.
• 🔑 Key Properties: It's a discrete distribution with only two possible outcomes: success (usually denoted as 1) or failure (usually denoted as 0).
• 🧮 Probability Mass Function (PMF): $P(X=x) = p^x (1-p)^{(1-x)}$, where $x$ is either 0 or 1, and $p$ is the probability of success.
• 📊 Expected Value (Mean): $E(X) = p$
• 📈 Variance: $Var(X) = p(1-p)$
• 🧪 Example: Flipping a coin once – Heads (success) or Tails (failure).

Practice Quiz

1. Question 1: What is the primary characteristic that defines a Bernoulli distribution?
   1. A) It models continuous data.
   2. B) It involves multiple independent trials.
   3. C) It models a single trial with two outcomes.
   4. D) It has infinite possible outcomes.
2. Question 2: If $p$ represents the probability of success in a Bernoulli trial, what does $(1-p)$ represent?
   1. A) The mean of the distribution.
   2. B) The variance of the distribution.
   3. C) The probability of failure.
   4. D) The standard deviation.
3. Question 3: For a Bernoulli random variable $X$, which of the following is the correct formula for the expected value, $E(X)$?
   1. A) $E(X) = 1 - p$
   2. B) $E(X) = p(1-p)$
   3. C) $E(X) = p$
   4. D) $E(X) = \frac{1}{p}$
4. Question 4: What is the variance of a Bernoulli distribution with success probability $p = 0.4$?
   1. A) 0.4
   2. B) 0.6
   3. C) 0.24
   4. D) 0.16
5. Question 5: In a Bernoulli trial, if success is defined as rolling a 6 on a fair die, what is the value of $p$?
   1. A) 1
   2. B) $\frac{1}{2}$
   3. C) $\frac{1}{6}$
   4. D) $\frac{5}{6}$
6. Question 6: Which of the following is NOT an example of a situation that can be modeled by a Bernoulli distribution?
   1. A) Whether a customer makes a purchase (yes/no).
   2. B) Measuring the height of students in a class.
   3. C) Whether a coin lands heads or tails.
   4. D) Whether a product passes or fails a quality check.
7. Question 7: A machine produces items, and each item has a probability of 0.05 of being defective. What is the probability that a randomly selected item is NOT defective?
   1. A) 0.05
   2. B) 0.95
   3. C) 1.00
   4. D) 0.00