Binomial Distribution and Its Properties

Introduction

Key Concepts

Definition of Binomial Distribution

The Binomial distribution represents the probability distribution of the number of successes in a sequence of $n$ independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: success or failure. The probability of success in each trial is denoted by $p$, and the probability of failure is $q = 1 - p$.

Conditions for Binomial Distribution

For a random variable $X$ to follow a Binomial distribution, the following four conditions must be satisfied:

• Fixed Number of Trials ($n$): The experiment consists of a fixed number of trials.
• Binary Outcomes: Each trial results in one of two outcomes: success or failure.
• Constant Probability ($p$): The probability of success, $p$, remains constant across all trials.
• Independence: The trials are independent; the outcome of one trial does not affect the others.

Probability Mass Function (PMF)

The PMF of a Binomially distributed random variable $X$ is given by:

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

where:

• $\binom{n}{k}$: Number of combinations of $n$ trials taken $k$ at a time.
• $p^{k}$: Probability of $k$ successes.
• $q^{n - k}$: Probability of $n - k$ failures.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function for the Binomial distribution is the probability that the variable $X$ is less than or equal to a certain value $k$:

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

Mean and Variance

The mean ($\mu$) and variance ($\sigma^2$) of a Binomial distribution are crucial for understanding its spread and central tendency.

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

where:

• $n$: Number of trials.
• $p$: Probability of success.
• $q$: Probability of failure ($q = 1 - p$).

Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance:

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

Examples of Binomial Distribution

Consider a scenario where a student is preparing for their IB Mathematics exam. Suppose the probability of correctly answering any given multiple-choice question (success) is $p = 0.7$, and there are $n = 10$ questions.

To find the probability that the student answers exactly $k = 7$ questions correctly:

$$ P(X = 7) = \binom{10}{7} (0.7)^{7} (0.3)^{3} \approx 0.2668 $$

This means there's approximately a 26.68% chance the student answers exactly 7 out of 10 questions correctly.

Applications of Binomial Distribution

Binomial distributions are widely applicable in various fields:

• Quality Control: Determining the probability of a certain number of defective items in a batch.
• Medicine: Estimating the effectiveness of a treatment where each patient either responds positively or negatively.
• Finance: Modeling the number of times a trader's strategy results in a profitable trade over a series of trades.
• Education: Analyzing the number of students who pass or fail an exam.

Manipulating the Binomial Formula

The Binomial coefficient, $\binom{n}{k}$, plays a pivotal role in the Binomial distribution. It calculates the number of ways to choose $k$ successes from $n$ trials:

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

For example, $\binom{5}{2} = 10$, meaning there are 10 ways to achieve 2 successes in 5 trials.

Relationship with Other Distributions

The Binomial distribution is related to several other probability distributions:

• Normal Distribution: For large $n$, the Binomial distribution approximates the Normal distribution with mean $np$ and variance $npq$.
• Poisson Distribution: When $n$ is large and $p$ is small, the Binomial distribution approaches the Poisson distribution with parameter $\lambda = np$.

Generating Functions

The generating function of the Binomial distribution is used to encapsulate the probabilities in a functional form, facilitating the computation of moments like the mean and variance:

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

Binomial Theorem

The Binomial Theorem is closely related and states that:

$$ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{k} b^{n - k} $$

This theorem underpins the Binomial distribution's PMF, where $a = p$ and $b = q$.

Real-World Problem Solving

Applying the Binomial distribution involves identifying whether the problem meets the necessary conditions and then using the PMF to calculate the desired probabilities. For instance:

Problem: A factory produces light bulbs with a 2% defect rate. If a random sample of 20 bulbs is selected, what is the probability that exactly 3 are defective?

Solution:

Here, $n = 20$, $p = 0.02$, and $k = 3$. Plugging into the PMF:

$$ P(X = 3) = \binom{20}{3} (0.02)^3 (0.98)^{17} \approx 0.000.136 $$

Thus, there's approximately a 0.0136% chance of finding exactly 3 defective bulbs in the sample.

Limitations of the Binomial Distribution

While powerful, the Binomial distribution has its limitations:

• Independence Assumption: Trials must be independent, which is not always the case in real-world scenarios.
• Fixed Probability: The probability of success must remain constant across trials, which may vary in dynamic environments.
• Discrete Outcomes: Only applicable to experiments with two possible outcomes (success or failure).

Comparison Table

Aspect	Binomial Distribution	Poisson Distribution
Number of Trials	Fixed ($n$)	Not fixed; counts events in a time interval
Probability of Success	Constant ($p$)	Assumed to be small
Type of Events	Independent Bernoulli trials	Independent events occurring randomly over time
PMF Formula	$\binom{n}{k} p^{k} q^{n - k}$	$\frac{\lambda^{k} e^{-\lambda}}{k!}$
Mean	$np$	$\lambda$
Variance	$npq$	$\lambda$

Summary and Key Takeaways