Statistics and Probability Cheat Sheet - NDA

1. NDA PYQ & Pattern Analysis

Statistics and Probability form one of the highest-weightage modules in the UPSC NDA Mathematics paper. Historically, candidate analysis reveals that 15 to 20 questions are directly asked from these topics, accounting for roughly 37.5 to 50 marks out of the 300 total marks in Paper 1.

The NTA/UPSC pattern emphasizes conceptual clarity over tedious computation. Questions frequently test Bayes' Theorem, properties of Binomial and Poisson distributions, moments (mean, variance, covariance), and standard rules of continuous probability distributions like the Normal and Exponential curves.

2. Foundations of Probability

Probability forms the bedrock of statistical decision-making. In NDA problems, axiomatic probability definitions and conditional reasoning are repeatedly tested.

Axioms of Probability

For any event \(A\) defined on a sample space \(\Omega\):

• Non-negativity & Boundedness: \(0 \le P(A) \le 1\)
• Total Probability of Sample Space: \(P(\Omega) = 1\)
• Countable Additivity: If \(A_1, A_2, \dots\) are mutually exclusive (disjoint) events, then \(P(\bigcup A_i) = \sum P(A_i)\).

Key Operational Rules

• Addition Rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
• Complementary Rule: \(P(A^c) = 1 - P(A)\)
• Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) provided \(P(B) > 0\)
• Multiplication Rule: \(P(A \cap B) = P(A|B)P(B)\)

Independence & Theorem Frameworks

Two events \(A\) and \(B\) are defined as independent if and only if \(P(A \cap B) = P(A)P(B)\).

Law of Total Probability: Given a partition of sample space into disjoint events \(A_1, A_2, \dots, A_n\):

Bayes' Theorem: Evaluates posterior probability given prior information:

3. Random Variables & Moments

A random variable maps outcomes of random processes to numerical values, quantified via distribution functions and central moments.

Cumulative Distribution Function (CDF)

The CDF for a random variable \(X\) is denoted by:

Expectation \(E[X]\)

Expectation represents the long-run average value of a random variable:

• Discrete Case: \(E[X] = \sum_x x P(X=x)\)
• Continuous Case: \(E[X] = \int_{-\infty}^{\infty} x f(x) dx\)
• Linearity Property: \(E[aX + bY + c] = aE[X] + bE[Y] + c\)

Variance, Covariance, and Correlation

Variance & Standard Deviation:

Covariance & Correlation Coefficient:

4. Discrete Probability Distributions

Bernoulli Distribution \(\text{Bernoulli}(p)\)

Models a single trial with success probability \(p\) where \(X \in \{0,1\}\):

• PMF: \(P(X=x) = p^x (1-p)^{1-x}\)
• Mean & Variance: \(E[X] = p\), \(\text{Var}(X) = p(1-p)\)

Binomial Distribution \(\text{Binomial}(n, p)\)

Models the total number of successes in \(n\) independent Bernoulli trials:

• PMF: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\)
• Mean & Variance: \(E[X] = np\), \(\text{Var}(X) = np(1-p)\)

Poisson Distribution \(\text{Poisson}(\lambda)\)

Represents event occurrences within a fixed interval given a constant rate \(\lambda\):

• PMF: \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\)
• Mean & Variance: \(E[X] = \lambda\), \(\text{Var}(X) = \lambda\)

Geometric Distribution \(\text{Geometric}(p)\)

Models the trial number on which the first success occurs:

• PMF: \(P(X=k) = (1-p)^{k-1}p\)
• Mean & Variance: \(E[X] = \frac{1}{p}\), \(\text{Var}(X) = \frac{1-p}{p^2}\)

5. Continuous Probability Distributions

Uniform Distribution \(\text{Uniform}(a, b)\)

• PDF: \(f(x) = \frac{1}{b-a}\) for \(x \in [a,b]\)
• Mean & Variance: \(E[X] = \frac{a+b}{2}\), \(\text{Var}(X) = \frac{(b-a)^2}{12}\)

Normal Distribution \(\mathcal{N}(\mu, \sigma^2)\)

• PDF: \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\bigl(-\frac{(x-\mu)^2}{2\sigma^2}\bigr)\)
• Standardization Transformation: \(Z = \frac{X-\mu}{\sigma} \sim \mathcal{N}(0,1)\)
• Empirical Rule: Approximately 68%, 95%, and 99.7% of values fall within \(\mu \pm 1\sigma\), \(\mu \pm 2\sigma\), and \(\mu \pm 3\sigma\) respectively.

Exponential Distribution \(\text{Exponential}(\lambda)\)

• PDF: \(f(x) = \lambda e^{-\lambda x}\) for \(x \ge 0\)

6. People Also Ask (FAQ)