Standard distributions - part1

Introduction

This tutorial provides a comprehensive overview of the binomial distribution, including its definition and derivation. Understanding the binomial distribution is essential for anyone studying statistics, as it is a foundational concept used in various fields including finance, research, and quality control.

Step 1: Understand the Definition of Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials. Each trial has two possible outcomes, often termed as "success" and "failure."

• Key Characteristics:
  • Fixed number of trials (n)
  • Two possible outcomes for each trial
  • Constant probability of success (p) across trials
  • Trials are independent

Step 2: Identify the Parameters of Binomial Distribution

To fully grasp the binomial distribution, it's crucial to identify its parameters.

• Parameters:
  • n: Total number of trials
  • p: Probability of success on each trial
  • q: Probability of failure (q = 1 - p)

Step 3: Write the Binomial Probability Formula

The probability of obtaining exactly k successes in n trials is given by the binomial probability formula:

[ P(X = k) = \binom{n}{k} p^k q^{n-k} ]

Where:

• ( \binom{n}{k} ) is the binomial coefficient, calculated as:

[ \binom{n}{k} = \frac{n!}{k!(n-k)!} ]

Step 4: Derive the Binomial Distribution Formula

To derive the binomial distribution formula, consider the following:

1. Choose k successes from the n trials. The number of ways to choose k successes is given by the binomial coefficient.
2. Calculate the probability of k successes and (n-k) failures:
   • Probability of k successes: ( p^k )
   • Probability of (n-k) failures: ( q^{n-k} )
3. Combine these elements using the multiplication rule of probability.

Step 5: Apply the Binomial Distribution

To see the binomial distribution in action, consider an example:

• Example Problem: A coin is flipped 10 times. What is the probability of getting exactly 6 heads (successes) if the probability of heads on a single flip is 0.5?

1. Define parameters:

   • n = 10 (trials)
   • k = 6 (successes)
   • p = 0.5 (probability of success)

2. Use the binomial formula:

[ P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^{10-6} ]

3. Calculate:

   • First, calculate the binomial coefficient ( \binom{10}{6} = 210 )
   • Then compute the probabilities: ( 0.5^6 ) and ( 0.5^4 )

4. Final Probability: [ P(X = 6) = 210 \cdot (0.5)^6 \cdot (0.5)^4 = 210 \cdot (0.5)^{10} = 210 \cdot \frac{1}{1024} \approx 0.205 ]

Conclusion

The binomial distribution is a powerful tool for modeling situations with a fixed number of trials and two outcomes. By understanding its definition, parameters, formula, and application, you can effectively analyze scenarios involving binomial experiments. For further exploration, consider studying the normal approximation of the binomial distribution for larger n or other related distributions such as the Poisson distribution.