Probability & Statistics with R - Complete Syllabus Notes (Basic to Advanced)

These notes are designed for your R-based Probability and Statistics syllabus and match the topics visible in your assessments.

0) Fast roadmap

1. Data basics and descriptive statistics
2. Probability fundamentals and random variables
3. Important distributions and R functions
4. Sampling distributions + CLT
5. Estimation and confidence intervals
6. Hypothesis testing (Z, t, two-sample, paired)
7. Chi-square tests (independence, goodness of fit)
8. ANOVA (one-way, randomized block, Latin square)
9. Decision guide: which test to use
10. Common mistakes and interpretation tips

1) Data basics and descriptive statistics

1.1 Data types

• Qualitative (categorical): gender, flavor, training program
• Quantitative discrete: number of defects, number of successes
• Quantitative continuous: height, weight, time, marks

1.2 Core summaries

For observations $x_1, x_2, ..., x_n$:

• Mean: $$ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i $$

• Sample variance: $$ s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 $$

• Sample standard deviation: $$ s = \sqrt{s^2} $$

• Coefficient of variation: $$ CV = \frac{s}{\bar{x}} \times 100% $$

1.3 R code template

# x is numeric vector
x <- c(10, 12, 15, 9, 14)

length(x)
mean(x)
median(x)
var(x)
sd(x)
IQR(x)
summary(x)

# frequency table for categorical data
major <- c("CSE", "ECE", "CSE", "ME", "ECE")
table(major)
prop.table(table(major))

# quick plots
hist(x, main = "Histogram", xlab = "x")
boxplot(x, main = "Boxplot")
barplot(table(major), main = "Category counts")

2) Probability fundamentals

2.1 Basic rules

• $0 \le P(A) \le 1$

• $P(S)=1$, $P(\emptyset)=0$

• Complement: $P(A^c)=1-P(A)$

• Addition rule: $$ P(A \cup B)=P(A)+P(B)-P(A\cap B) $$

• Conditional probability: $$ P(A\mid B)=\frac{P(A\cap B)}{P(B)}, \quad P(B)>0 $$

• Multiplication rule: $$ P(A\cap B)=P(A\mid B)P(B) $$

• Independence: $A,B$ independent if $$ P(A\cap B)=P(A)P(B) $$

• Bayes theorem: $$ P(A_i\mid B)=\frac{P(B\mid A_i)P(A_i)}{\sum_j P(B\mid A_j)P(A_j)} $$

2.2 Random variables

• Discrete RV: PMF $p(x)=P(X=x)$
• Continuous RV: PDF $f(x)$, with $P(a\le X\le b)=\int_a^b f(x),dx$
• CDF: $F(x)=P(X\le x)$

2.3 Expectation and variance

• Discrete: $$ E(X)=\sum x p(x), \quad Var(X)=E(X^2)-[E(X)]^2 $$

• Linear transform $Y=aX+b$: $$ E(Y)=aE(X)+b, \quad Var(Y)=a^2Var(X) $$

3) Important distributions + R commands

R naming pattern:

• d... density/pmf
• p... cumulative probability
• q... quantile
• r... random generation

3.1 Binomial: $X\sim Bin(n,p)$

• PMF: $$ P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}, \quad x=0,1,...,n $$
• Mean $np$, variance $np(1-p)$

# P(X = 3) when n=10, p=0.2
dbinom(3, size = 10, prob = 0.2)

# P(X <= 3)
pbinom(3, size = 10, prob = 0.2)

# Simulate
rbinom(100, size = 10, prob = 0.2)

3.2 Poisson: $X\sim Pois(\lambda)$

• PMF: $$ P(X=x)=e^{-\lambda}\frac{\lambda^x}{x!} $$
• Mean = variance = $\lambda$

dpois(4, lambda = 2.5)
ppois(4, lambda = 2.5)
rpois(100, lambda = 2.5)

3.3 Normal: $X\sim N(\mu,\sigma^2)$

• Standardization: $$ Z=\frac{X-\mu}{\sigma} $$

# P(X <= 68) for N(65, 3^2)
pnorm(68, mean = 65, sd = 3)

# x such that P(X <= x)=0.95
qnorm(0.95, mean = 65, sd = 3)

# Generate values
rnorm(100, mean = 65, sd = 3)

3.4 t distribution

Used when population sigma unknown (especially small samples).

# CDF and quantile
t_prob <- pt(2.1, df = 14)
t_crit <- qt(0.975, df = 14)

3.5 Chi-square distribution

If $Z_i\sim N(0,1)$ independent, then $$ \chi^2 = \sum_{i=1}^{k} Z_i^2 \sim \chi_k^2 $$

pchisq(10.2, df = 6)
qchisq(0.95, df = 6)

3.6 F distribution

Ratio of scaled chi-square variables; appears in ANOVA.

pf(3.2, df1 = 2, df2 = 18)
qf(0.95, df1 = 2, df2 = 18)

4) Sampling distributions and CLT

4.1 Mean of sample mean

If population mean is $\mu$ and variance is $\sigma^2$: $$ E(\bar{X})=\mu, \quad Var(\bar{X})=\frac{\sigma^2}{n} $$

Standard error: $$ SE(\bar{X})=\frac{\sigma}{\sqrt{n}} \quad (\text{or } s/\sqrt{n}) $$

4.2 Central Limit Theorem (CLT)

For large $n$, sample mean is approximately normal: $$ \bar{X} \approx N\left(\mu, \frac{\sigma^2}{n}\right) $$

Useful for problems like: $P(a < \bar{X} < b)$, $P(\bar{X}>c)$ in your assessments.

# Example: P(160 < Xbar < 165), mu=162, sigma=12, n=36
mu <- 162
sigma <- 12
n <- 36
se <- sigma / sqrt(n)
pnorm(165, mu, se) - pnorm(160, mu, se)

5) Estimation and confidence intervals

5.1 CI for mean (sigma known)

$$ \bar{x} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}} $$

5.2 CI for mean (sigma unknown)

$$ \bar{x} \pm t_{\alpha/2, n-1}\frac{s}{\sqrt{n}} $$

5.3 CI for difference of means

• Known sigmas / large-sample Z: $$ (\bar{x}_1-\bar{x}2) \pm z{\alpha/2}\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} $$

• Unknown unequal variances: Welch t interval (t.test(..., var.equal=FALSE))

# One-sample t CI
x <- c(10, 11, 9, 13, 12, 10)
t.test(x, conf.level = 0.95)

# Two-sample Welch CI
a <- c(56, 60, 58, 61, 57)
b <- c(52, 49, 55, 50, 53)
t.test(a, b, var.equal = FALSE, conf.level = 0.95)

6) Hypothesis testing

General steps:

1. Define $H_0$ and $H_1$
2. Choose significance level $\alpha$ (commonly 0.05)
3. Compute test statistic
4. Obtain p-value or critical-value decision
5. Conclude in context

6.1 One-sample Z test (large sample or known sigma)

$$ Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} $$

# Manual Z test
xbar <- 815
mu0 <- 800
sigma <- 90
n <- 64
z <- (xbar - mu0) / (sigma / sqrt(n))
p_value_right <- 1 - pnorm(z)
z; p_value_right

6.2 One-sample t test

$$ t = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}, \quad df=n-1 $$

x <- c(48, 52, 50, 47, 51, 49, 53)
t.test(x, mu = 50, alternative = "two.sided")

6.3 Two-sample Z test (known sigmas)

$$ Z = \frac{(\bar{X}_1-\bar{X}_2)-\Delta_0}{\sqrt{\sigma_1^2/n_1 + \sigma_2^2/n_2}} $$

xbar1 <- 72; xbar2 <- 68
sigma1 <- 10; sigma2 <- 12
n1 <- 40; n2 <- 36
delta0 <- 0
z <- ((xbar1 - xbar2) - delta0) / sqrt(sigma1^2/n1 + sigma2^2/n2)
p_two <- 2 * (1 - pnorm(abs(z)))
z; p_two

6.4 Two-sample t test (unknown variances)

• Welch default in R:

t.test(group1, group2, var.equal = FALSE)

• Equal variances assumed (pooled):

t.test(group1, group2, var.equal = TRUE)

6.5 Paired t test

For paired differences $d_i = x_i - y_i$: $$ t = \frac{\bar{d}}{s_d/\sqrt{n}}, \quad df=n-1 $$

before <- c(95, 102, 98, 100, 97)
after  <- c(99, 105, 100, 103, 101)
t.test(after, before, paired = TRUE, alternative = "greater")

6.6 Type I and Type II errors

• Type I: reject true $H_0$ (probability = $\alpha$)
• Type II: fail to reject false $H_0$ (probability = $\beta$)
• Power = $1-\beta$

7) Chi-square methods

7.1 Chi-square test of independence (contingency table)

Used for association between two categorical variables.

$$ \chi^2 = \sum \frac{(O_{ij}-E_{ij})^2}{E_{ij}}, \quad E_{ij}=\frac{(\text{row total})(\text{col total})}{N} $$

Degrees of freedom: $$ (r-1)(c-1) $$

tbl <- matrix(c(20, 30, 25, 35), nrow = 2, byrow = TRUE)
chisq.test(tbl, correct = FALSE)

7.2 Goodness-of-fit (GOF)

Check whether observed counts follow a specified distribution.

$$ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $$

obs <- c(18, 32, 30, 20)
# expected probabilities (must sum to 1)
p <- c(0.2, 0.3, 0.3, 0.2)
chisq.test(obs, p = p)

7.3 Fitting binomial then GOF

For grouped counts over x=0,1,...,n:

1. Estimate $\hat p = \frac{\sum x f_x}{n\sum f_x}$
2. Compute expected counts: $E_x = N\cdot P(X=x; n,\hat p)$
3. Run chi-square GOF (pool tiny expected counts if needed)

x <- 0:4
f <- c(8, 22, 30, 25, 15)  # observed frequencies
N <- sum(f)
n <- 4
p_hat <- sum(x * f) / (n * N)
exp_counts <- N * dbinom(x, size = n, prob = p_hat)

# Manual chi-square statistic
chi_sq <- sum((f - exp_counts)^2 / exp_counts)
chi_sq

# Approximate p-value with df = k - 1 - m
# k = number of categories, m = parameters estimated (here m=1 for p)
df <- length(x) - 1 - 1
p_value <- 1 - pchisq(chi_sq, df = df)
p_value

8) ANOVA and experimental designs

8.1 Why ANOVA

Use when comparing 3 or more means without doing many t-tests.

Hypotheses:

• $H_0$: all group means equal
• $H_1$: at least one mean differs

F statistic: $$ F = \frac{MS_{between}}{MS_{within}} $$

8.2 One-way ANOVA

Model: $$ Y_{ij} = \mu + \tau_i + \epsilon_{ij} $$

Assumptions:

• independent observations
• normal residuals
• equal variances

# df has columns: score, group
fit <- aov(score ~ group, data = df)
summary(fit)

# Diagnostics
par(mfrow = c(2,2))
plot(fit)

# Post-hoc if significant
TukeyHSD(fit)

8.3 Randomized block / two-factor additive model

Used when one nuisance factor (block/machine/day) should be controlled.

Model: $$ Y_{ij} = \mu + \tau_i + \beta_j + \epsilon_{ij} $$

# Example with treatment and block
fit_block <- aov(response ~ treatment + block, data = df)
summary(fit_block)

8.4 Latin Square Design (LSD)

Controls two blocking factors (rows and columns) plus treatment.

Model: $$ Y_{ijk} = \mu + \tau_i + \rho_j + \kappa_k + \epsilon_{ijk} $$

# Columns: yield, treatment, row, col
fit_lsd <- aov(yield ~ treatment + row + col, data = df)
summary(fit_lsd)

8.5 Interpreting ANOVA output

• If Pr(>F) < alpha: reject $H_0$ for that factor
• Report in context: "There is significant evidence that mean output differs by machine."

9) Practical R workflow for exam problems

9.1 Reusable function for quick summary

quick_stats <- function(x) {
  c(
    n = length(x),
    mean = mean(x),
    sd = sd(x),
    var = var(x),
    min = min(x),
    q1 = quantile(x, 0.25),
    median = median(x),
    q3 = quantile(x, 0.75),
    max = max(x)
  )
}

9.2 p-value to decision helper

decide <- function(p, alpha = 0.05) {
  if (p < alpha) "Reject H0" else "Fail to reject H0"
}

9.3 Normality and variance checks

# For ANOVA / t assumptions
shapiro.test(residuals(fit))

# Equal variances across groups (base R)
bartlett.test(score ~ group, data = df)

10) Which test should I use? (decision guide)

1. One sample mean vs target:

• sigma known / large n: one-sample Z
• sigma unknown: one-sample t

1. Two independent means:

• sigmas known: two-sample Z
• unknown sigmas: Welch t (default)

1. Same subjects before-after:

• paired t-test

1. Two categorical variables association:

• chi-square independence

1. Observed counts vs theoretical distribution:

• chi-square GOF

1. Compare 3+ group means:

• one-way ANOVA

1. Compare treatments with one block factor:

• randomized block ANOVA

1. Treatments with two block factors:

• Latin square ANOVA

11) Common mistakes to avoid

• Mixing up independent and paired tests
• Using Z test when sigma is unknown and sample is small
• Ignoring one-sided vs two-sided alternative
• Declaring "accept H0" (better: "fail to reject H0")
• Running many t-tests instead of ANOVA for 3+ groups
• Not checking expected counts for chi-square tests
• Forgetting to interpret results in domain context

12) High-yield exam templates

Template A: One-sample Z test

xbar <- 0; mu0 <- 0; sigma <- 1; n <- 1
z <- (xbar - mu0) / (sigma / sqrt(n))
p <- 2 * (1 - pnorm(abs(z)))
list(z = z, p_value = p)

Template B: Two-sample t test (Welch)

t.test(x, y, alternative = "two.sided", var.equal = FALSE, conf.level = 0.95)

Template C: Paired t test

t.test(after, before, paired = TRUE, alternative = "greater")

Template D: Chi-square independence

tab <- table(df$A, df$B)
chisq.test(tab, correct = FALSE)

Template E: One-way ANOVA + Tukey

fit <- aov(y ~ grp, data = df)
summary(fit)
TukeyHSD(fit)

Template F: Block + treatment ANOVA

fit <- aov(y ~ treatment + block, data = df)
summary(fit)

Template G: Latin square

fit <- aov(y ~ treatment + row + col, data = df)
summary(fit)

13) Final revision checklist

1. Can you identify variable type and suitable test quickly?
2. Can you write $H_0$, $H_1$ correctly (with direction)?
3. Do you know the exact R command and main output line to report?
4. Can you interpret p-value in one sentence with context?
5. Can you state assumptions and whether they are reasonable?

If you can do these five consistently, you are exam-ready for most Prob-Stats practical questions in R.

14) Topic-wise practice questions with answers (easy to advanced)

Use these as self-tests. Try each question first, then compare with the answer.

14.1 Descriptive statistics and data handling

Easy Q1: Given x <- c(5, 7, 7, 9, 12), find mean and median.

Answer:

• Mean = $(5+7+7+9+12)/5 = 8$
• Median = 7

R check:

x <- c(5, 7, 7, 9, 12)
mean(x)
median(x)

Medium Q2: For x <- c(10, 12, 8, 9, 11, 10), find sample variance and SD.

Answer:

• Use $s^2 = \frac{1}{n-1}\sum (x_i-\bar{x})^2$
• Numerically: $s^2 = 2$, so $s = \sqrt{2} \approx 1.414$

R check:

x <- c(10, 12, 8, 9, 11, 10)
var(x)
sd(x)

Advanced Q3: Create a summary table for a student score vector containing n, mean, sd, min, Q1, median, Q3, max.

Answer:

scores <- c(68, 72, 75, 80, 81, 85, 90, 92)
c(
  n = length(scores),
  mean = mean(scores),
  sd = sd(scores),
  min = min(scores),
  q1 = quantile(scores, 0.25),
  median = median(scores),
  q3 = quantile(scores, 0.75),
  max = max(scores)
)

14.2 Probability fundamentals

Easy Q1: If $P(A)=0.4$, find $P(A^c)$.

Answer:

• $P(A^c)=1-0.4=0.6$

Medium Q2: If $P(A)=0.5$, $P(B)=0.6$, and $P(A\cap B)=0.3$, find $P(A\cup B)$.

Answer:

• $P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.5+0.6-0.3=0.8$

Advanced Q3 (Bayes): A machine comes from Plant 1 (60%) or Plant 2 (40%). Defect rates are 2% and 5% respectively. If an item is defective, find probability it came from Plant 2.

Answer: $$ P(P2\mid D)=\frac{P(D\mid P2)P(P2)}{P(D\mid P1)P(P1)+P(D\mid P2)P(P2)} =\frac{0.05\cdot0.4}{0.02\cdot0.6+0.05\cdot0.4} =\frac{0.02}{0.032}=0.625 $$

14.3 Distributions

Easy Q1 (Binomial): For $X\sim Bin(5, 0.2)$, find $P(X=1)$.

Answer: $$ P(X=1)=\binom{5}{1}(0.2)^1(0.8)^4=0.4096 $$

R check:

dbinom(1, size = 5, prob = 0.2)

Medium Q2 (Poisson): If $X\sim Pois(3)$, find $P(X\le2)$.

Answer:

ppois(2, lambda = 3)

Advanced Q3 (Normal): If $X\sim N(70, 8^2)$, find $P(65 < X < 82)$.

Answer:

pnorm(82, mean = 70, sd = 8) - pnorm(65, mean = 70, sd = 8)

14.4 Sampling distribution and CLT

Easy Q1: Population has $\mu=50$, $\sigma=10$, sample size $n=25$. Find standard error of $\bar{X}$.

Answer: $$ SE=\frac{\sigma}{\sqrt{n}}=\frac{10}{5}=2 $$

Medium Q2: With same parameters, find $P(\bar{X}>53)$.

Answer: $$ Z=\frac{53-50}{2}=1.5,\quad P(\bar{X}>53)=1-\Phi(1.5)\approx0.0668 $$

R check:

1 - pnorm(53, mean = 50, sd = 2)

Advanced Q3: For $\mu=200$, $\sigma=30$, find n such that $SE\le 3$.

Answer: $$ \frac{30}{\sqrt{n}}\le 3 \Rightarrow \sqrt{n}\ge10 \Rightarrow n\ge100 $$

14.5 Confidence intervals

Easy Q1: Given $\bar{x}=40$, $\sigma=12$, $n=36$, 95% CI for mean.

Answer: $$ 40 \pm 1.96\cdot\frac{12}{6}=40\pm3.92=(36.08, 43.92) $$

Medium Q2: Sample x <- c(21, 20, 19, 24, 22, 23, 20, 21); find 95% t-CI.

Answer:

x <- c(21, 20, 19, 24, 22, 23, 20, 21)
t.test(x, conf.level = 0.95)$conf.int

Advanced Q3: Interpret a 95% CI (18.2, 22.4).

Answer:

• We are 95% confident that the true population mean lies between 18.2 and 22.4.
• Since it excludes 20? No, 20 is inside; so evidence is not strong against $\mu=20$ at 5% level.

14.6 Hypothesis testing (Z, t, paired)

Easy Q1 (one-sample t): Test $H_0: \mu=50$ vs $H_1: \mu\ne50$ for x <- c(49, 52, 47, 51, 50, 53, 48) at $\alpha=0.05$.

Answer:

x <- c(49, 52, 47, 51, 50, 53, 48)
t.test(x, mu = 50, alternative = "two.sided")

Decision rule: reject $H_0$ if p-value < 0.05.

Medium Q2 (two-sample Welch): Compare two teaching methods: a <- c(72, 75, 78, 74, 77) b <- c(69, 71, 73, 70, 72)

Answer:

a <- c(72, 75, 78, 74, 77)
b <- c(69, 71, 73, 70, 72)
t.test(a, b, var.equal = FALSE)

Advanced Q3 (paired): Before and after training scores: before <- c(60, 62, 58, 65, 61, 59) after <- c(64, 63, 61, 68, 65, 62) Test if training improved scores.

Answer:

before <- c(60, 62, 58, 65, 61, 59)
after <- c(64, 63, 61, 68, 65, 62)
t.test(after, before, paired = TRUE, alternative = "greater")

14.7 Chi-square tests

Easy Q1 (independence): Use table: $$ \begin{bmatrix} 18 & 22\ 25 & 35 \end{bmatrix} $$ Is there association?

Answer:

tab <- matrix(c(18, 22, 25, 35), nrow = 2, byrow = TRUE)
chisq.test(tab, correct = FALSE)

Medium Q2 (GOF): Observed counts: c(30, 45, 25) Expected probabilities: c(0.3, 0.4, 0.3)

Answer:

obs <- c(30, 45, 25)
chisq.test(obs, p = c(0.3, 0.4, 0.3))

Advanced Q3 (binomial fit + GOF): Given counts for x=0..3 as f <- c(12, 30, 38, 20) where $n=3$ per trial, fit binomial and test GOF.

Answer:

x <- 0:3
f <- c(12, 30, 38, 20)
N <- sum(f)
n <- 3
p_hat <- sum(x * f) / (n * N)
exp_counts <- N * dbinom(x, size = n, prob = p_hat)
chi_sq <- sum((f - exp_counts)^2 / exp_counts)
df <- length(x) - 1 - 1
p_value <- 1 - pchisq(chi_sq, df)
list(p_hat = p_hat, chi_sq = chi_sq, p_value = p_value)

14.8 ANOVA and design of experiments

Easy Q1 (one-way ANOVA): Check if 3 machines have equal means.

Answer:

df <- data.frame(
  y = c(21, 22, 20, 25, 24, 23, 28, 27, 29),
  machine = factor(rep(c("M1", "M2", "M3"), each = 3))
)
fit <- aov(y ~ machine, data = df)
summary(fit)

Medium Q2 (post-hoc): After significant ANOVA, identify which pairs differ.

Answer:

TukeyHSD(fit)

Advanced Q3 (block + treatment): Analyze treatment effect controlling for block.

Answer:

df2 <- data.frame(
  response = c(15, 18, 16, 17, 20, 19, 14, 16, 15, 18, 21, 19),
  treatment = factor(rep(c("T1", "T2", "T3"), times = 4)),
  block = factor(rep(c("B1", "B2", "B3", "B4"), each = 3))
)
fit_block <- aov(response ~ treatment + block, data = df2)
summary(fit_block)

Advanced Q4 (Latin square): Fit a Latin square model with row and column blocks.

Answer:

fit_lsd <- aov(yield ~ treatment + row + col, data = lsd_df)
summary(fit_lsd)

14.9 Mixed challenge set (exam-style)

Q1: A sample of size 64 has mean 103. Population sigma is 16. Test $H_0: \mu=100$ vs $H_1: \mu>100$.

Answer: $$ Z=\frac{103-100}{16/8}=1.5,\quad p=1-\Phi(1.5)\approx0.0668 $$ At $\alpha=0.05$: fail to reject $H_0$.

Q2: You have before/after values on the same 12 people. Which test?

Answer:

• Paired t-test (t.test(after, before, paired=TRUE)).

Q3: You have 4 training programs and want to compare mean output.

Answer:

• One-way ANOVA (aov(output ~ program, data = df)).

Q4: Gender and flavor preference are both categorical. Which test?

Answer:

• Chi-square test of independence.

15) How to use this practice bank

1. Solve all easy questions first without seeing answers.
2. Move to medium only after scoring at least 80% on easy.
3. For advanced questions, write full interpretation sentence with p-value.
4. Re-run only incorrect questions after 24 hours (spaced revision).
5. Before exam, solve section 14.9 in one sitting under time limit.