Binomial Distribution and Its Properties

Introduction

Key Concepts

Definition of Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. A Bernoulli trial is an experiment with exactly two possible outcomes: success (with probability $p$) and failure (with probability $q = 1 - p$).

The probability mass function (PMF) of the binomial distribution is given by:

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

where:

• $n$ = number of trials
• $k$ = number of successes
• $\binom{n}{k}$ = binomial coefficient, calculated as $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• $p$ = probability of success on a single trial
• $q$ = probability of failure on a single trial

Assumptions of the Binomial Distribution

The binomial distribution relies on four key assumptions:

1. There are a fixed number of trials ($n$).
2. Each trial is independent of the others.
3. Each trial has only two possible outcomes: success or failure.
4. The probability of success ($p$) remains constant across trials.

Mean and Variance

The mean ($\mu$) and variance ($\sigma^2$) of a binomial distribution are essential properties that describe its central tendency and dispersion:

$$\mu = np$$ $$\sigma^2 = npq$$

These formulas indicate that the mean increases linearly with the number of trials and the probability of success, while the variance depends on both the probability of success and failure.

Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance and provides a measure of the spread of the distribution:

$$\sigma = \sqrt{npq}$$

Probability Mass Function (PMF)

The PMF gives the probability of obtaining exactly $k$ successes in $n$ trials:

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

This function is fundamental for calculating probabilities associated with specific numbers of successes.

Cumulative Distribution Function (CDF)

The CDF of the binomial distribution calculates the probability of obtaining up to $k$ successes:

$$P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i} p^i q^{n-i}$$

Mode of the Binomial Distribution

The mode is the most probable number of successes. It can be approximated by:

$$\text{Mode} \approx \lfloor (n+1)p \rfloor$$

where $\lfloor \cdot \rfloor$ denotes the floor function.

Skewness

Skewness measures the asymmetry of the distribution. For the binomial distribution:

$$\text{Skewness} = \frac{q - p}{\sqrt{npq}}$$

Positive skewness indicates a distribution tailing to the right, while negative skewness indicates a tailing to the left.

Kurtosis

Kurtosis measures the "tailedness" of the distribution:

$$\text{Kurtosis} = \frac{1 - 6pq}{npq}$$

Higher kurtosis indicates more weight in the tails.

Examples

Example 1: A true-false quiz consists of 10 questions. What is the probability of getting exactly 7 questions correct by guessing?

Here, $n = 10$, $k = 7$, $p = 0.5$, and $q = 0.5$.

Applying the PMF:

$$P(X = 7) = \binom{10}{7} (0.5)^7 (0.5)^3 = 120 \times 0.0078125 \times 0.125 = 0.1171875$$

Example 2: A factory produces light bulbs with a defect rate of 2%. What is the probability that in a random sample of 100 bulbs, exactly 3 are defective?

Here, $n = 100$, $k = 3$, $p = 0.02$, and $q = 0.98$.

Applying the PMF:

$$P(X = 3) = \binom{100}{3} (0.02)^3 (0.98)^{97} \approx 161700 \times 0.000008 \times 0.132 \approx 0.170$$

Advanced Concepts

Binomial Theorem and Distribution

The binomial theorem provides a connection between algebra and probability. It states that:

$$ (p + q)^n = \sum_{k=0}^{n} \binom{n}{k} p^k q^{n-k} $$

This expansion mirrors the binomial distribution's PMF, indicating that the probabilities sum to 1: $$ \sum_{k=0}^{n} P(X = k) = 1 $$

Derivation of Mean and Variance

To derive the mean ($\mu$) and variance ($\sigma^2$) of the binomial distribution, consider the expectation and the expectation of the square:

Mean:

$$\mu = E(X) = \sum_{k=0}^{n} k \cdot P(X = k)$$

By simplifying, we find: $$\mu = np$$

Variance:

$$\sigma^2 = E(X^2) - [E(X)]^2$$

Through calculation, this results in: $$\sigma^2 = npq$$

Generating Functions

The probability generating function (PGF) for the binomial distribution is a powerful tool for deriving moments and other properties:

$$G_X(t) = (q + pt)^n$$

Expanding this function provides coefficients corresponding to the probabilities $P(X = k)$.

Moment Generating Function (MGF)

The MGF is given by: $$M_X(t) = \left( q + pe^t \right)^n$$

Using the MGF, one can derive the moments of the distribution by taking derivatives with respect to $t$.

Approximation to Normal Distribution

For large $n$, the binomial distribution can be approximated by a normal distribution with mean $\mu = np$ and variance $\sigma^2 = npq$. This approximation, known as the De Moivre-Laplace theorem, simplifies calculations for probabilities involving large sample sizes.

Conditions for Normal Approximation:

• $n$ is large
• Both $np \geq 5$ and $nq \geq 5$

Applications in Hypothesis Testing

The binomial distribution is instrumental in hypothesis testing, particularly in scenarios involving proportions. For instance, determining whether a new drug has a significantly different success rate compared to an existing treatment involves binomial probability calculations.

Interdisciplinary Connections

The binomial distribution extends its utility beyond mathematics into fields such as genetics, where it models the inheritance of traits, and quality control in manufacturing, where it assesses defect rates. In finance, it aids in portfolio risk assessment by evaluating the number of profitable assets within a portfolio.

Complex Problem-Solving

Problem: A basketball player has a free-throw success rate of 80%. If the player takes 15 free throws, what is the probability of making at least 12?

Solution: We need to calculate $P(X \geq 12)$ where $X \sim \text{Binomial}(15, 0.8)$. This is equivalent to:

$$P(X \geq 12) = \sum_{k=12}^{15} \binom{15}{k} (0.8)^k (0.2)^{15-k}$$

Calculating each term and summing them gives the desired probability.

Using Binomial Distribution in Real-World Data Analysis

In clinical trials, determining the number of patients responding to a treatment among a fixed sample size can be modeled using the binomial distribution. This facilitates the assessment of treatment efficacy and statistical significance.

Comparison Table

Aspect	Binomial Distribution	Normal Distribution
Type	Discrete	Continuous
Parameters	Number of trials ($n$), Probability of success ($p$)	Mean ($\mu$), Standard deviation ($\sigma$)
Support	$k = 0, 1, 2, \dots, n$	All real numbers
Shape	Depends on $p$ and $n$; can be symmetric or skewed	Symmetrical, bell-shaped curve
Applicability	Independent trials with two outcomes	Large sample sizes, continuous data
Mean	$np$	$\mu$
Variance	$npq$	$\sigma^2$

Summary and Key Takeaways