📊 R – Normal & Binomial Distribution Explained with Code Examples

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🧲 Introduction – Statistical Distributions in R

Understanding probability distributions is essential in statistics and data science. R provides built-in functions to simulate, analyze, and visualize popular distributions.

This guide focuses on:

• 📈 Normal Distribution (continuous)
• 🧮 Binomial Distribution (discrete)

🎯 In this guide, you’ll learn:

• How to generate random numbers from distributions in R
• Plot Probability Density Functions (PDFs) and Cumulative Distributions (CDFs)
• Understand syntax of dnorm(), rbinom(), etc.
• Apply these distributions to real-world examples

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🔹 1. Normal Distribution in R

A normal distribution (bell curve) is defined by:

• mean: center of the distribution
• sd: spread (standard deviation)

✅ Generate Normally Distributed Data

x <- rnorm(1000, mean = 50, sd = 10)
hist(x, breaks = 20, col = "lightblue", main = "Histogram of Normal Distribution")

🔍 Explanation:

• rnorm(n, mean, sd) generates n values from a normal distribution

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✅ Plot Normal Curve

x <- seq(-3, 3, by = 0.1)
y <- dnorm(x, mean = 0, sd = 1)
plot(x, y, type = "l", col = "blue", main = "Standard Normal Distribution")

🔍 Explanation:

• dnorm(x, mean, sd) gives density values
• type = "l" draws a line

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✅ Cumulative Probability

pnorm(1.96, mean = 0, sd = 1)   # Output: ~0.975

🧠 Meaning: Probability that a standard normal value is ≤ 1.96 is ~97.5%

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✅ Inverse Lookup (Quantile Function)

qnorm(0.975, mean = 0, sd = 1)   # Output: ~1.96

🧠 Meaning: The 97.5th percentile in standard normal distribution is 1.96

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🔹 2. Binomial Distribution in R

A binomial distribution models:

• n independent trials
• each trial has success probability (p)
• output = number of successes

✅ Simulate Binomial Outcomes

rbinom(10, size = 5, prob = 0.5)

🔍 Explanation:

• rbinom(n, size, prob) generates n samples where each sample is the number of successes in size trials with probability prob

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✅ Probability Mass Function (PMF)

x <- 0:10
y <- dbinom(x, size = 10, prob = 0.5)
barplot(y, names.arg = x, col = "orange", main = "Binomial PMF (n=10, p=0.5)")

🔍 Explanation:

• dbinom(x, size, prob) gives the probability of x successes

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✅ Cumulative Probability

pbinom(6, size = 10, prob = 0.5)   # P(X ≤ 6)

✅ Quantile (Inverse)

qbinom(0.8, size = 10, prob = 0.5)

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📊 Distribution Function Overview Table

Function	Purpose	Normal Example	Binomial Example
r*()	Generate random values	rnorm()	rbinom()
d*()	PDF / PMF	dnorm()	dbinom()
p*()	Cumulative probability (CDF)	pnorm()	pbinom()
q*()	Quantile function (inverse CDF)	qnorm()	qbinom()

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📌 Summary – Recap & Next Steps

R provides intuitive functions to work with distributions both analytically and visually.
Understanding these is vital for hypothesis testing, simulation, and data modeling.

🔍 Key Takeaways:

• Use r*, d*, p*, and q* functions for distributions
• Normal: continuous, symmetric, bell curve
• Binomial: discrete, success/failure outcomes
• Visualize distributions with hist(), barplot(), plot()

⚙️ Real-World Relevance:
Used in probability models, quality control, A/B testing, risk analysis, genetics, and machine learning.

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❓ FAQs – Distributions in R

❓ What is the difference between dnorm() and pnorm()?
✅ dnorm() gives density (height of the curve); pnorm() gives cumulative probability.

❓ How to simulate 100 coin tosses in R?
✅ Use:

rbinom(100, size = 1, prob = 0.5)

❓ What does qnorm(0.975) return?
✅ The value of Z such that 97.5% of values lie below it (≈ 1.96).

❓ When to use binomial vs normal distribution?
✅ Binomial is for discrete successes in finite trials; Normal is for continuous, bell-shaped data.

❓ Can I plot both distributions on the same graph?
✅ Yes, but normalize the binomial data for proper comparison.

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