3.4 Binomial Probability Disrubution

Summary of the Binomial Probability Distribution

The binomial probability distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is one of the most commonly used distributions in statistics and probability theory. Below are the key concepts:

1. Binomial Experiment

A binomial experiment is a statistical experiment that satisfies the following conditions:

1. Fixed number of trials (n): The experiment consists of a fixed number of trials, denoted by ( n ).

2. Independent trials: Each trial is independent of the others. The outcome of one trial does not affect the outcome of another.

3. Two possible outcomes: Each trial has only two possible outcomes, often referred to as success (with probability ( p )) and failure (with probability ( q = 1 - p )).

4. Constant probability of success (p): The probability of success, ( p ), remains the same for each trial.

2. Binomial Random Variable

The binomial random variable, denoted by ( X ), represents the number of successes in ( n ) trials of a binomial experiment. It can take integer values from ( 0 ) to ( n ).

3. Probability Mass Function (PMF)

The probability mass function of the binomial distribution gives the probability of obtaining exactly ( k ) successes in ( n ) trials. It is given by: [ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ] where:

• ( \binom{n}{k} ) is the binomial coefficient, calculated as: [ \binom{n}{k} = \frac{n!}{k!(n-k)!} ]

• ( p ) is the probability of success on a single trial.

• ( k ) is the number of successes (( k = 0, 1, 2, \dots, n )).

4. Parameters of the Binomial Distribution

The binomial distribution is characterized by two parameters:

1. Number of trials (n): The fixed number of independent trials.

2. Probability of success (p): The probability of success on a single trial.

5. Mean and Variance

The mean (expected value) and variance of a binomial random variable ( X ) are given by:

• Mean: [ \mu = E(X) = np ]

• Variance: [ \sigma^2 = V(X) = np(1-p) ]

• Standard Deviation: [ \sigma = \sqrt{np(1-p)} ]

6. Applications

The binomial distribution is used in situations where there are a fixed number of independent trials, each with the same probability of success. Examples include:

• Counting the number of defective items in a batch of products.

• Determining the number of heads in a series of coin flips.

• Calculating the probability of a certain number of successes in a survey or experiment.

7. Shape of the Binomial Distribution

• The shape of the binomial distribution depends on the values of ( n ) and ( p ).

• For a fixed ( n ), the distribution is symmetric when ( p = 0.5 ), skewed right when ( p < 0.5 ), and skewed left when ( p > 0.5 ).

• As ( n ) increases, the binomial distribution approaches a normal distribution (this is known as the Central Limit Theorem).

8. Cumulative Probability

The cumulative probability of a binomial random variable ( X ) is the probability that ( X ) takes a value less than or equal to a specific value ( k ). It is calculated as: [ P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{n-i} ]

9. Assumptions

• The trials are independent.

• The probability of success ( p ) is constant across trials.

• The number of trials ( n ) is fixed in advance.

Key Takeaways

• The binomial distribution is a powerful tool for modeling the number of successes in a fixed number of independent trials.

• It is defined by two parameters: ( n ) (number of trials) and ( p ) (probability of success).

• The mean and variance of the distribution are ( np ) and ( np(1-p) ), respectively.

• It is widely used in real-world applications involving binary outcomes.