IIT-JEE MAINS 2027 cheat sheet for Statistics and Probability! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

\n \n Statistics & Probability: Essential Formula Cheat Sheet

A rigorous summary of high-yield definitions, distribution metrics, and probability theorems tailored for IIT-JEE Main preparation.

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1. Statistics: Measures of Central Tendency & Dispersion

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Statistics evaluates the properties of datasets through central values and variance metrics.

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Mean and Variance of Grouped/Ungrouped Data

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For a discrete frequency distribution with values \(x_i\) and frequencies \(f_i\), the mean \(\bar{x}\) is given by:

\n \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\]\n \n

The variance \(\sigma^2\) measures the spread of data points around the mean:

\n \[\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} = \frac{\sum f_i x_i^2}{\sum f_i} - \bar{x}^2\]\n \n

Standard deviation \(\sigma\) is the non-negative square root of variance, given by \(\sigma = \sqrt{\sigma^2}\).

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Properties of Variance

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• Linear Transformation: If each observation \(x_i\) is transformed to \(y_i = ax_i + b\), then the new mean is \(\bar{y} = a\bar{x} + b\) and the new variance becomes \(\sigma_y^2 = a^2 \sigma_x^2\).
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• Combined Variance: For two groups with sizes \(n_1, n_2\), means \(\bar{x}_1, \bar{x}_2\), and variances \(\sigma_1^2, \sigma_2^2\), the combined variance \(\sigma^2\) satisfies:\n \[\sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}\]\n where \(d_1 = \bar{x}_1 - \bar{x}_{combined}\) and \(d_2 = \bar{x}_2 - \bar{x}_{combined}\).\n
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2. Fundamentals of Probability Theory

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Probability quantifies the likelihood of occurrence of events within a sample space \(S\).

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Axioms & Addition Theorem

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For any two events \(A\) and \(B\):

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

If \(A\) and \(B\) are mutually exclusive, \(P(A \cap B) = 0\), reducing the relation to \(P(A \cup B) = P(A) + P(B)\).

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Conditional Probability & Independence

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The conditional probability of event \(A\) given that event \(B\) has occurred is defined as:

\n \[P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0\]\n

Events \(A\) and \(B\) are independent if and only if \(P(A \cap B) = P(A) \cdot P(B)\), which implies \(P(A|B) = P(A)\).

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3. Advanced Theorems & Distributions

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Total Probability & Bayes' Theorem

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Let \(E_1, E_2, \dots, E_n\) be mutually exclusive and exhaustive events forming a partition of \(S\). For any event \(A\):

\n \[P(A) = \sum_{i=1}^{n} P(E_i) \cdot P(A|E_i)\]\n

By Bayes' Theorem, the posterior probability of partition event \(E_k\) given event \(A\) is:

\n \[P(E_k|A) = \frac{P(E_k) \cdot P(A|E_k)}{\sum_{i=1}^{n} P(E_i) \cdot P(A|E_i)}\]\n\n

Binomial Distribution

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A random variable \(X\) following a Binomial Distribution \(B(n, p)\) with \(n\) independent trials and success probability \(p\) (where \(q = 1 - p\)) has probability mass function:

\n \[P(X = r) = \binom{n}{r} p^r q^{n-r}, \quad r \in \{0, 1, 2, \dots, n\}\]\n

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• Mean: \(\mu = np\)
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• Variance: \(\sigma^2 = npq\) (Note that \(\sigma^2 < \mu\) always holds for a Binomial distribution).
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