Chapter 2: Probability & Distributions

Probability theory, common distributions, and their applications in statistical analysis using R.

Author

Pawan Jangra

Published

January 20, 2025

Learning Objectives: Understand core probability concepts, work with common distributions (Normal, Binomial, Poisson, t), and apply these in real-world contexts using R’s built-in distribution functions.

1 Probability Fundamentals

Every distribution in R has four functions:

Prefix	Function	Returns
`d`	`dnorm(x)`	Probability density at x
`p`	`pnorm(q)`	P(X ≤ q) — CDF
`q`	`qnorm(p)`	Quantile (inverse CDF)
`r`	`rnorm(n)`	n random samples

Code

library(tidyverse)
library(knitr)

2 The Normal Distribution

The Normal distribution is the cornerstone of statistical inference — thanks to the Central Limit Theorem.

Code

# Compare Normal distributions with different parameters
x_vals <- seq(-5, 15, length.out = 500)

normal_df <- bind_rows(
  data.frame(x=x_vals, density=dnorm(x_vals, mean=0, sd=1),   label="N(0,1)"),
  data.frame(x=x_vals, density=dnorm(x_vals, mean=3, sd=1.5), label="N(3,1.5)"),
  data.frame(x=x_vals, density=dnorm(x_vals, mean=5, sd=0.5), label="N(5,0.5)"),
  data.frame(x=x_vals, density=dnorm(x_vals, mean=2, sd=2.5), label="N(2,2.5)")
)

ggplot(normal_df, aes(x=x, y=density, color=label)) +
  geom_line(linewidth=1.2) +
  scale_color_manual(values=c("#2563eb","#7c3aed","#0d9488","#d97706")) +
  labs(title="Normal Distributions with Different Parameters",
       subtitle="Location (μ) shifts the center; Scale (σ) controls spread",
       x="x", y="Density", color="Distribution") +
  theme_minimal(base_size=11) +
  theme(legend.position="right", panel.grid.minor=element_blank())

2.1 The 68-95-99.7 Rule

Code

x <- seq(-4, 4, length.out=1000)
df_norm <- data.frame(x=x, y=dnorm(x))

ggplot(df_norm, aes(x,y)) +
  geom_area(data=filter(df_norm, x >= -3 & x <= 3), fill="#93c5fd", alpha=0.3) +
  geom_area(data=filter(df_norm, x >= -2 & x <= 2), fill="#60a5fa", alpha=0.4) +
  geom_area(data=filter(df_norm, x >= -1 & x <= 1), fill="#2563eb", alpha=0.5) +
  geom_line(linewidth=1.5, color="#1e3a8a") +
  geom_vline(xintercept=c(-3,-2,-1,0,1,2,3), linetype="dashed", 
             alpha=0.4, color="#64748b") +
  annotate("text", x=0, y=0.2, label="68%\n(±1σ)", size=3.5, fontface="bold", color="white") +
  annotate("text", x=1.7, y=0.05, label="95%\n(±2σ)", size=3, color="#1e40af") +
  annotate("text", x=2.7, y=0.015, label="99.7%\n(±3σ)", size=2.5, color="#1e3a8a") +
  labs(title="Empirical Rule: 68-95-99.7",
       x="Standard Deviations from Mean", y="Density") +
  theme_minimal(base_size=11) +
  theme(panel.grid.minor=element_blank())

Code

# Probability calculations
cat("P(Z ≤ 1.96)  =", round(pnorm(1.96), 4), "\n")

P(Z ≤ 1.96)  = 0.975

Code

cat("P(-1.96 ≤ Z ≤ 1.96) =", round(pnorm(1.96) - pnorm(-1.96), 4), "\n")

P(-1.96 ≤ Z ≤ 1.96) = 0.95

Code

cat("P(Z > 2.33)  =", round(1 - pnorm(2.33), 4), "\n")

P(Z > 2.33)  = 0.0099

Code

cat("95th percentile (z):", round(qnorm(0.95), 4), "\n")

95th percentile (z): 1.6449

Code

cat("99th percentile (z):", round(qnorm(0.99), 4), "\n")

99th percentile (z): 2.3263

3 Binomial Distribution

Used when we have n independent trials, each with probability p of success.

Application: In a survey of 100 households, if 35% own agricultural land, what’s the probability that exactly 40 own land?

Code

n_trials <- 100
p_success <- 0.35

binom_df <- data.frame(
  k = 0:n_trials,
  prob = dbinom(0:n_trials, n_trials, p_success)
) |> filter(prob > 0.0001)

# Highlight specific outcomes
binom_df <- binom_df |>
  mutate(highlight = case_when(
    k == 40 ~ "k = 40",
    k >= 35 & k <= 40 ~ "35-40",
    TRUE ~ "Other"
  ))

ggplot(binom_df, aes(x=k, y=prob, fill=highlight)) +
  geom_col(width=0.7) +
  scale_fill_manual(values=c("k = 40"="#e11d48","35-40"="#7c3aed","Other"="#93c5fd"),
                    name="") +
  labs(title=paste0("Binomial(n=", n_trials, ", p=", p_success, ")"),
       x="Number of successes (k)", y="Probability P(X = k)") +
  theme_minimal(base_size=11) +
  theme(legend.position="top", panel.grid.minor=element_blank())

Code

cat("P(X = 40) =", round(dbinom(40, 100, 0.35), 4))

P(X = 40) = 0.0474

Code

cat("\nP(X ≥ 40) =", round(1 - pbinom(39, 100, 0.35), 4))


P(X ≥ 40) = 0.1724

Code

cat("\nExpected value E(X) = np =", 100*0.35)


Expected value E(X) = np = 35

Code

cat("\nSD(X) = √(np(1-p)) =", round(sqrt(100*0.35*0.65), 3))


SD(X) = √(np(1-p)) = 4.77

4 Poisson Distribution

Used for count data — events happening in a fixed interval.

Application: If a field surveyor visits an average of 8 households per hour, what’s the probability of visiting exactly 10 in an hour?

Code

lambdas <- c(1, 3, 8, 15)
pois_df <- expand.grid(
  k = 0:30,
  lambda = lambdas
) |>
  mutate(
    prob  = dpois(k, lambda),
    label = paste0("λ = ", lambda)
  ) |> filter(prob > 0.001)

ggplot(pois_df, aes(x=k, y=prob, color=label)) +
  geom_line(linewidth=1.1) +
  geom_point(size=2) +
  scale_color_manual(values=c("#2563eb","#7c3aed","#0d9488","#d97706")) +
  labs(title="Poisson Distributions",
       x="k (count)", y="P(X = k)", color="Parameter") +
  theme_minimal(base_size=11) +
  theme(legend.position="right", panel.grid.minor=element_blank())

5 The t-Distribution

Used when the population standard deviation is unknown (which is almost always in practice).

Code

x <- seq(-5, 5, length.out=500)
t_df <- bind_rows(
  data.frame(x=x, y=dnorm(x), dist="Normal (Z)"),
  data.frame(x=x, y=dt(x, df=3),  dist="t(df=3)"),
  data.frame(x=x, y=dt(x, df=10), dist="t(df=10)"),
  data.frame(x=x, y=dt(x, df=30), dist="t(df=30)")
)

ggplot(t_df, aes(x, y, color=dist, linetype=dist)) +
  geom_line(linewidth=1.1) +
  scale_color_manual(values=c("#0f172a","#e11d48","#7c3aed","#2563eb")) +
  scale_linetype_manual(values=c("dashed","solid","solid","solid")) +
  labs(title="t-Distribution vs Normal",
       subtitle="As df increases, t approaches the Standard Normal",
       x="x", y="Density", color="Distribution", linetype="Distribution") +
  theme_minimal(base_size=11) +
  theme(legend.position="right", panel.grid.minor=element_blank())

Code

# Critical values
cat("t-critical values (two-tailed, α=0.05):\n")

t-critical values (two-tailed, α=0.05):

Code

for (df in c(5, 10, 20, 30, 60, 120)) {
  cat(sprintf("  df=%3d: t = %.4f\n", df, qt(0.975, df)))
}

  df=  5: t = 2.5706
  df= 10: t = 2.2281
  df= 20: t = 2.0860
  df= 30: t = 2.0423
  df= 60: t = 2.0003
  df=120: t = 1.9799

Code

cat(sprintf("  Normal: z = %.4f\n", qnorm(0.975)))

  Normal: z = 1.9600

6 Practice Exercises

Try It Yourself

Normal: If IQ scores are N(100, 15²), find P(IQ > 130) and the 90th percentile
Binomial: In 200 trials with p=0.25, find the mean, SD, and P(X ≥ 60)
Poisson: If λ=5 calls/hour, find P(0 calls) and P(X ≥ 8)
t: For a sample of n=25, find the 95% CI for the mean wage from Chapter 1’s data

Next: Chapter 3 — Regression Analysis →

--- title: "Chapter 2: Probability & Distributions" description: "Probability theory, common distributions, and their applications in statistical analysis using R." author: "Pawan Jangra" date: "2025-01-20" toc: true number-sections: true --- > **Learning Objectives:** Understand core probability concepts, work with common distributions (Normal, Binomial, Poisson, t), and apply these in real-world contexts using R's built-in distribution functions. ## Probability Fundamentals Every distribution in R has four functions: | Prefix | Function | Returns | |--------|----------|---------| | `d` | `dnorm(x)` | Probability density at x | | `p` | `pnorm(q)` | P(X ≤ q) — CDF | | `q` | `qnorm(p)` | Quantile (inverse CDF) | | `r` | `rnorm(n)` | n random samples | ```{r setup} #| message: false library(tidyverse) library(knitr) ``` ## The Normal Distribution The Normal distribution is the cornerstone of statistical inference — thanks to the Central Limit Theorem. ```{r normal-dist} #| fig-height: 5 # Compare Normal distributions with different parameters x_vals <- seq(-5, 15, length.out = 500) normal_df <- bind_rows( data.frame(x=x_vals, density=dnorm(x_vals, mean=0, sd=1), label="N(0,1)"), data.frame(x=x_vals, density=dnorm(x_vals, mean=3, sd=1.5), label="N(3,1.5)"), data.frame(x=x_vals, density=dnorm(x_vals, mean=5, sd=0.5), label="N(5,0.5)"), data.frame(x=x_vals, density=dnorm(x_vals, mean=2, sd=2.5), label="N(2,2.5)") ) ggplot(normal_df, aes(x=x, y=density, color=label)) + geom_line(linewidth=1.2) + scale_color_manual(values=c("#2563eb","#7c3aed","#0d9488","#d97706")) + labs(title="Normal Distributions with Different Parameters", subtitle="Location (μ) shifts the center; Scale (σ) controls spread", x="x", y="Density", color="Distribution") + theme_minimal(base_size=11) + theme(legend.position="right", panel.grid.minor=element_blank()) ``` ### The 68-95-99.7 Rule ```{r empirical-rule} #| fig-height: 4 x <- seq(-4, 4, length.out=1000) df_norm <- data.frame(x=x, y=dnorm(x)) ggplot(df_norm, aes(x,y)) + geom_area(data=filter(df_norm, x >= -3 & x <= 3), fill="#93c5fd", alpha=0.3) + geom_area(data=filter(df_norm, x >= -2 & x <= 2), fill="#60a5fa", alpha=0.4) + geom_area(data=filter(df_norm, x >= -1 & x <= 1), fill="#2563eb", alpha=0.5) + geom_line(linewidth=1.5, color="#1e3a8a") + geom_vline(xintercept=c(-3,-2,-1,0,1,2,3), linetype="dashed", alpha=0.4, color="#64748b") + annotate("text", x=0, y=0.2, label="68%\n(±1σ)", size=3.5, fontface="bold", color="white") + annotate("text", x=1.7, y=0.05, label="95%\n(±2σ)", size=3, color="#1e40af") + annotate("text", x=2.7, y=0.015, label="99.7%\n(±3σ)", size=2.5, color="#1e3a8a") + labs(title="Empirical Rule: 68-95-99.7", x="Standard Deviations from Mean", y="Density") + theme_minimal(base_size=11) + theme(panel.grid.minor=element_blank()) ``` ```{r prob-calculations} # Probability calculations cat("P(Z ≤ 1.96) =", round(pnorm(1.96), 4), "\n") cat("P(-1.96 ≤ Z ≤ 1.96) =", round(pnorm(1.96) - pnorm(-1.96), 4), "\n") cat("P(Z > 2.33) =", round(1 - pnorm(2.33), 4), "\n") cat("95th percentile (z):", round(qnorm(0.95), 4), "\n") cat("99th percentile (z):", round(qnorm(0.99), 4), "\n") ``` ## Binomial Distribution Used when we have **n independent trials**, each with probability **p** of success. **Application:** In a survey of 100 households, if 35% own agricultural land, what's the probability that exactly 40 own land? ```{r binomial} #| fig-height: 4 n_trials <- 100 p_success <- 0.35 binom_df <- data.frame( k = 0:n_trials, prob = dbinom(0:n_trials, n_trials, p_success) ) |> filter(prob > 0.0001) # Highlight specific outcomes binom_df <- binom_df |> mutate(highlight = case_when( k == 40 ~ "k = 40", k >= 35 & k <= 40 ~ "35-40", TRUE ~ "Other" )) ggplot(binom_df, aes(x=k, y=prob, fill=highlight)) + geom_col(width=0.7) + scale_fill_manual(values=c("k = 40"="#e11d48","35-40"="#7c3aed","Other"="#93c5fd"), name="") + labs(title=paste0("Binomial(n=", n_trials, ", p=", p_success, ")"), x="Number of successes (k)", y="Probability P(X = k)") + theme_minimal(base_size=11) + theme(legend.position="top", panel.grid.minor=element_blank()) cat("P(X = 40) =", round(dbinom(40, 100, 0.35), 4)) cat("\nP(X ≥ 40) =", round(1 - pbinom(39, 100, 0.35), 4)) cat("\nExpected value E(X) = np =", 100*0.35) cat("\nSD(X) = √(np(1-p)) =", round(sqrt(100*0.35*0.65), 3)) ``` ## Poisson Distribution Used for **count data** — events happening in a fixed interval. **Application:** If a field surveyor visits an average of 8 households per hour, what's the probability of visiting exactly 10 in an hour? ```{r poisson} #| fig-height: 4 lambdas <- c(1, 3, 8, 15) pois_df <- expand.grid( k = 0:30, lambda = lambdas ) |> mutate( prob = dpois(k, lambda), label = paste0("λ = ", lambda) ) |> filter(prob > 0.001) ggplot(pois_df, aes(x=k, y=prob, color=label)) + geom_line(linewidth=1.1) + geom_point(size=2) + scale_color_manual(values=c("#2563eb","#7c3aed","#0d9488","#d97706")) + labs(title="Poisson Distributions", x="k (count)", y="P(X = k)", color="Parameter") + theme_minimal(base_size=11) + theme(legend.position="right", panel.grid.minor=element_blank()) ``` ## The t-Distribution Used when the **population standard deviation is unknown** (which is almost always in practice). ```{r t-dist} #| fig-height: 4 x <- seq(-5, 5, length.out=500) t_df <- bind_rows( data.frame(x=x, y=dnorm(x), dist="Normal (Z)"), data.frame(x=x, y=dt(x, df=3), dist="t(df=3)"), data.frame(x=x, y=dt(x, df=10), dist="t(df=10)"), data.frame(x=x, y=dt(x, df=30), dist="t(df=30)") ) ggplot(t_df, aes(x, y, color=dist, linetype=dist)) + geom_line(linewidth=1.1) + scale_color_manual(values=c("#0f172a","#e11d48","#7c3aed","#2563eb")) + scale_linetype_manual(values=c("dashed","solid","solid","solid")) + labs(title="t-Distribution vs Normal", subtitle="As df increases, t approaches the Standard Normal", x="x", y="Density", color="Distribution", linetype="Distribution") + theme_minimal(base_size=11) + theme(legend.position="right", panel.grid.minor=element_blank()) ``` ```{r t-critical} # Critical values cat("t-critical values (two-tailed, α=0.05):\n") for (df in c(5, 10, 20, 30, 60, 120)) { cat(sprintf(" df=%3d: t = %.4f\n", df, qt(0.975, df))) } cat(sprintf(" Normal: z = %.4f\n", qnorm(0.975))) ``` ## Practice Exercises ::: {.callout-tip} ## Try It Yourself 1. **Normal:** If IQ scores are N(100, 15²), find P(IQ > 130) and the 90th percentile 2. **Binomial:** In 200 trials with p=0.25, find the mean, SD, and P(X ≥ 60) 3. **Poisson:** If λ=5 calls/hour, find P(0 calls) and P(X ≥ 8) 4. **t:** For a sample of n=25, find the 95% CI for the mean wage from Chapter 1's data ::: **Next:** [Chapter 3 — Regression Analysis →](chapter3.html)