GATE EE

Probability and Statistics Notes for GATE Electrical Engineering (EE)

Lecture Notes

SEC 01

Sampling Theorems

1Sampling Theorems
1Central Limit Theorem (CLT)

For large sample size \(n\), sample mean \(\bar{X}\) follows normal distribution:

\[\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)\]
1Law of Large Numbers

As \(n \to \infty\): \(\bar{X} \to \mu\) (sample mean converges to population mean)

1Standard Error
\[SE = \frac{\sigma}{\sqrt{n}}\]
where \(\sigma\) is population standard deviation
1Key Point

CLT applies when \(n \geq 30\) regardless of population distribution

SEC 02

Conditional Probability

1Conditional Probability
1Definition

Probability of event A given event B has occurred:

\[P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0\]
1Bayes’ Theorem
\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]
1Total Probability
\[P(B) = \sum_{i=1}^{n} P(B|A_i) \cdot P(A_i)\]
1Independence

Events A and B are independent if: \(P(A|B) = P(A)\) or \(P(A \cap B) = P(A) \cdot P(B)\)

SEC 03

Measures of Central Tendency

1Mean, Median, Mode
1Arithmetic Mean
\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}\]
1Median
1Mode

Most frequently occurring value in the dataset

1Relationship

For skewed distributions: Mean \(\neq\) Median \(\neq\) Mode

SEC 04

Standard Deviation

1Standard Deviation & Variance
1Population Variance
\[\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2\]
1Sample Variance
\[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2\]
1Standard Deviation
\[\sigma = \sqrt{\sigma^2}, \quad s = \sqrt{s^2}\]
1Alternative Formula
\[\sigma^2 = E[X^2] - (E[X])^2\]
\[s^2 = \frac{\sum x_i^2 - n\bar{x}^2}{n-1}\]
SEC 05

Random Variables

1Random Variables
1Definition

Random Variable (RV): Function that assigns numerical values to outcomes of random experiment

1Expected Value
\[E[X] = \sum_{i} x_i P(X = x_i) \quad \text{(Discrete)}\]
\[E[X] = \int_{-\infty}^{\infty} x f(x) dx \quad \text{(Continuous)}\]
1Variance
\[Var(X) = E[X^2] - (E[X])^2\]
\[Var(aX + b) = a^2 Var(X)\]
1Properties
SEC 06

Discrete Distributions

1Discrete Distributions
1Probability Mass Function (PMF)
\[P(X = k), \quad \sum_k P(X = k) = 1\]
1Cumulative Distribution Function (CDF)
\[F(x) = P(X \leq x) = \sum_{k \leq x} P(X = k)\]
1Common Discrete Distributions
SEC 07

Continuous Distributions

1Continuous Distributions
1Probability Density Function (PDF)
\[f(x) \geq 0, \quad \int_{-\infty}^{\infty} f(x) dx = 1\]
\[P(a < X < b) = \int_a^b f(x) dx\]
1Cumulative Distribution Function (CDF)
\[F(x) = P(X \leq x) = \int_{-\infty}^x f(t) dt\]
\[f(x) = \frac{dF(x)}{dx}\]
1Common Continuous Distributions
SEC 08

Poisson Distribution

1Poisson Distribution
1Definition

Models number of events in fixed interval: \(X \sim \text{Pois}(\lambda)\)

1PMF
\[P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}, \quad k = 0, 1, 2, \ldots\]
1Parameters
1Applications
SEC 09

Normal Distribution

1Normal Distribution
1Definition

\(X \sim N(\mu, \sigma^2)\) - Bell-shaped curve

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

\(Z \sim N(0,1)\): \(Z = \frac{X - \mu}{\sigma}\)

\[\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}\]
1Properties
SEC 10

Binomial Distribution

1Binomial Distribution
1Definition

\(n\) independent trials, each with success probability \(p\): \(X \sim \text{Bin}(n,p)\)

1PMF
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n\]
where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
1Parameters
1Normal Approximation

When \(np \geq 5\) and \(n(1-p) \geq 5\): \(X \approx N(np, np(1-p))\)

SEC 11

Correlation Analysis

1Correlation Analysis
1Pearson Correlation Coefficient
\[r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2 \sum_{i=1}^{n}(y_i - \bar{y})^2}}\]
Alternative: \(r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\)
1Properties
1Interpretation
SEC 12

Regression Analysis

1Regression Analysis
1Simple Linear Regression
\[y = a + bx + \epsilon\]
where \(y\) is dependent, \(x\) is independent variable
1Least Squares Estimates
\[b = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2} = \frac{S_{xy}}{S_{xx}}\]
\[a = \bar{y} - b\bar{x}\]
1Coefficient of Determination
\[R^2 = r^2 = \frac{\text{Explained Variation}}{\text{Total Variation}}\]
\(R^2\) represents proportion of variance explained by regression
1Key Point

Regression line always passes through \((\bar{x}, \bar{y})\)

SEC 13

Important Formulas Summary

1Quick Formula Reference
1Distribution Parameters
Distribution Mean Variance
Binomial \(np\) \(np(1-p)\)
Poisson \(\lambda\) \(\lambda\)
Normal \(\mu\) \(\sigma^2\)
Uniform \(\frac{a+b}{2}\) \(\frac{(b-a)^2}{12}\)
1Key Relationships