Probability and Statistics

Sheet 1

Probability & Distributions

10 formulas

Key formulas

Union of Events ☆

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Conditional Probability ☆

\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]

Bayes' Theorem ☆

\[ P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)} \]

Law of Total Probability ☆

\[ P(B) = \sum_i P(B \mid A_i)\,P(A_i) \]

the denominator Bayes' theorem needs

Expectation & Variance ☆

\[ E[X] = \sum x\,P(x) \qquad \text{Var}(X) = E[X^2] - (E[X])^2 \]

Binomial Distribution ☆

\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]

mean = np, variance = np(1−p)

Poisson Distribution ☆

\[ P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!} \]

mean = variance = λ

Normal (Gaussian) PDF ☆

\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

Standardization (z-score) ☆

\[ z = \frac{x - \mu}{\sigma} \]

Exponential Distribution ☆

\[ f(x) = \lambda e^{-\lambda x} \]

mean = 1/λ

Test your recall

Can you recall the union of events formula?

Reveal formula

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

GATE traps 3 traps

Mean = Variance is a Poisson-only fact

Only the Poisson distribution has mean equal to variance (both λ) -- don't carry that assumption into Binomial (mean np, variance np(1-p)) or Normal questions.

Bayes' denominator needs every path to B

P(B) in Bayes' theorem must be expanded via the law of total probability over every way B can occur -- forgetting a branch is the single most common Bayes mistake.

Independent vs. mutually exclusive are opposite extremes

Two events with non-zero probability can't be both independent and mutually exclusive -- mutually exclusive events are maximally dependent (one happening rules out the other).

Sheet 2

Statistical Inference

6 formulas

Key formulas

Sample Mean & Variance ☆

\[ \bar{x} = \frac{\sum x}{n} \qquad s^2 = \frac{\sum (x-\bar{x})^2}{n-1} \]

Central Limit Theorem ☆

\[ \bar{X} \sim N\!\left(\mu, \frac{\sigma^2}{n}\right) \]

for large n, whatever the population's own distribution

Confidence Interval (mean, known σ) ☆

\[ \bar{x} \pm z^{*} \frac{\sigma}{\sqrt{n}} \]

t-statistic ☆

\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]

use when σ is unknown / n is small

Chi-Squared Statistic ☆

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

Correlation Coefficient ☆

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

always between −1 and +1

Test your recall

Can you recall the sample mean & variance formula?

Reveal formula

\[ \bar{x} = \frac{\sum x}{n} \qquad s^2 = \frac{\sum (x-\bar{x})^2}{n-1} \]

GATE traps 3 traps

t, not z, for small samples with unknown σ

Use the t-distribution whenever the population standard deviation is unknown -- the z-test assumes σ is known, which is rare in practice and a common trap when n is small.

Correlation only captures linear relationships

r = 0 doesn't mean no relationship -- only no LINEAR relationship. A strong non-linear pattern (e.g. y = x^2 over symmetric x) can still give r ≈ 0.

Correlation is not causation

A high |r| shows association, never causation -- a classic GATE distractor pairs a correct correlation calculation with an incorrect causal conclusion.