<h2>GATE Data Science and AI Probability and Statistics Syllabus</h2>

**Counting (permutation and combinations), probability axioms, Sample space, events, independent events, mutually exclusive events, marginal, conditional and joint probability, Bayes Theorem, conditional expectation and variance, mean, median, mode and standard deviation, correlation, and covariance, random variables, discrete random variables and probability mass functions, uniform, Bernoulli, binomial distribution, Continuous random variables and probability distribution function, uniform, exponential, Poisson, normal, standard normal, t-distribution, chi-squared distributions, cumulative distribution function, Conditional PDF, Central limit theorem, confidence interval, z-test, t-test, chi-squared test.**

Here’s an overview of GATE Data Science and AI Probability and Statistics Syllabus

**Counting (Permutations and Combinations)**: Permutations refer to the arrangements of objects where the order matters, while combinations refer to selections of objects where the order doesn’t matter.**Probability Axioms**: The three basic probability axioms are:- Non-negativity: Probabilities are non-negative.
- Normalization: The probability of the entire sample space is 1.
- Additivity: The probability of the union of mutually exclusive events is the sum of their individual probabilities.

**Sample Space, Events**: The sample space is the set of all possible outcomes of an experiment, while events are subsets of the sample space.**Independent Events**: Events are independent if the occurrence of one event does not affect the probability of the occurrence of the other.**Mutually Exclusive Events**: Mutually exclusive events cannot occur simultaneously.**Marginal, Conditional, and Joint Probability**: Marginal probability refers to the probability of an event occurring without any conditions. Conditional probability is the probability of an event occurring given that another event has already occurred. Joint probability is the probability of the intersection of two events occurring together.**Bayes’ Theorem**: Bayes’ Theorem is a formula used to revise the probability of an event based on new evidence.**Conditional Expectation and Variance**: Conditional expectation and variance are measures of the expected value and variance of a random variable given certain conditions.**Descriptive Statistics (Mean, Median, Mode, Standard Deviation)**: These are measures used to describe the central tendency and dispersion of data.**Correlation and Covariance**: Correlation measures the strength and direction of the relationship between two variables, while covariance measures the extent to which two variables change together.**Random Variables**: Random variables are variables that take on numerical values as a result of a random experiment.**Discrete Random Variables and Probability Mass Functions**: Discrete random variables have countable outcomes, and their probabilities are described by probability mass functions (PMFs).**Continuous Random Variables and Probability Distribution Functions**: Continuous random variables have uncountable outcomes, and their probabilities are described by probability distribution functions (PDFs).**Common Probability Distributions**: This includes uniform, exponential, Poisson, normal, standard normal, t-distribution, and chi-squared distributions.**Cumulative Distribution Function (CDF)**: The CDF gives the probability that a random variable is less than or equal to a certain value.**Central Limit Theorem**: The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the distribution of the population.**Inferential Statistics (Confidence Interval, z-test, t-test, chi-squared test)**: These are techniques used to make inferences or decisions about populations based on sample data.

Understanding these concepts is essential for analyzing data, making predictions, and drawing conclusions in various fields, including data science and artificial intelligence.

**GATE DA Subject wise syllabus:**

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