ThinkLab

Learning Objectives

Understand packages and libraries in R and how they work.
Learn basics of ggplot2.
Calculate and visualize probabilities from:
- Binomial distribution.
- Normal distribution.

Key Functions

ggplot()
dbinom()
pbinom()
geom_segment()
pnorm()
stat_function()
dnorm()
geom_area()

Introduction to ggplot2

ggplot2 is a powerful visualization package based on the Grammar of Graphics.
Three key components in ggplot():
1. Data – the dataset.
2. Aesthetics (aes) – variables mapped to axes, color, size, etc.
3. Geometric object – the type of plot (geom_histogram, geom_point, etc.).

ggplot Basics Example

Dataset: penguins.csv.

Start with data:
```
ggplot(data = penguins)
```

Add aesthetics:

ggplot(data = penguins, aes(x = body_mass_g))

Add geometry:

ggplot(data = penguins, aes(x = body_mass_g)) +
  geom_histogram()

Mnemonic: Data → Aesthetics → Geometric object.

Binomial Distribution in R

Probability Mass Function

dbinom(x, size, prob) → P(X = x)
- x: number of successes.
- size: number of trials.
- prob: probability of success.

Example: exactly 3 heads in 4 flips, p=0.5.

dbinom(x = 3, size = 4, prob = 0.5)

Cumulative Distribution

pbinom(q, size, prob, lower.tail) → P(X ≤ q) if lower.tail=TRUE.
Default: lower.tail = TRUE.
- TRUE → P(X ≤ x).
- FALSE → P(X > x).

Example: at most 1 head in 4 flips.

pbinom(q = 1, size = 4, prob = 0.5)

Binomial Visualization

Create a data frame with outcomes and probabilities:

df1 <- data.frame(x = 0:4,
                  y = dbinom(0:4, size = 4, prob = 0.5))

Plot with geom_segment:

ggplot(df1, aes(x = x, xend = x, y = 0, yend = y)) +
  geom_segment() +
  labs(title = "Binomial(4, 0.5)",
       x = "Number of Heads",
       y = "Probability")

Reminder: Always add informative title + axis labels.

Normal Distribution in R

Cumulative Probabilities

Function: pnorm(q, mean, sd, lower.tail).
Example: P(X < 50) when mean=80, sd=15.
```
pnorm(q = 50, mean = 80, sd = 15)
```
For the standard normal distribution (Z): mean=0, sd=1 (default).

Visualizing Normal PDF

Use stat_function() with dnorm():

ggplot(data.frame(x = c(-4, 4)), aes(x = x)) +
  stat_function(fun = dnorm, args = list(0, 1))

Improved with labels and formatting:

ggplot(data.frame(x = c(-4, 4)), aes(x = x)) +
  stat_function(fun = dnorm, args = list(0, 1),
                col = "black", lwd = 1) +
  labs(title = "Normal(0, 1)", x = "", y = "Density")

Shading Areas in Normal Distribution

Use geom_area() to shade regions.

Example: Shade region where Z ≤ 1.4.

ggplot(data.frame(x = c(-4, 4)), aes(x = x)) +
  geom_area(stat = "function", fun = dnorm,
            args = list(0, 1),
            fill = "lightblue", xlim = c(-4, 1.4)) +
  stat_function(fun = dnorm, args = list(0, 1),
                col = "black", lwd = 1) +
  labs(title = "Normal(0, 1)", y = "Density")

Key: specify xlim for shaded area bounds.

Uniform Distribution in R

Function: punif(q, min, max, lower.tail).

Example: Wait time between 20–70 minutes. P(X ≥ 30).

punif(q = 30, min = 20, max = 70, lower.tail = FALSE)

General Reminders

Always install once, load per session for packages.
Binomial distribution: use dbinom() for exact probabilities, pbinom() for cumulative.
Normal distribution: use pnorm() for probabilities, dnorm() for density plotting.
Uniform distribution: use punif().
Visualizations should always include:
- Title.
- Clear axis labels.
- Professional style suitable for reports.

应用统计学方法（二）