GATE DA & AI Engineering Probability and Statistics Syllabus

Probability and Statistics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Probability and Statistics in GATE DA & AI Engineering, not a summary of it.

1. 1

   Counting

   10 topics

   • 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-Distributions
     • Chi-Squared Distributions
     • Cumulative Distribution Function
     • Conditional PDF
     • Central Limit Theorem
     • Confidence Interval
     • z-Test
     • t-Test
     • Chi-Squared Test

Probability and Statistics flashcards for GATE DA & AI Engineering

18 of 50 cards from the Probability and Statistics deck — real questions with worked answers.

1. What is the number of permutations of $n$ distinct objects taken $r$ at a time?

   $$P(n,r) = \frac{n!}{(n-r)!}$$

2. What is the number of combinations of $n$ distinct objects taken $r$ at a time?

   $$C(n,r) = \binom{n}{r} = \frac{n!}{r!\,(n-r)!}$$

3. How do permutations and combinations differ?

   Permutations count arrangements where order matters; combinations count selections where order does not matter. They are related by $P(n,r) = r!\cdot C(n,r)$.

4. State the three Kolmogorov axioms of probability.

   For event $A$: (1) $P(A) \geq 0$; (2) $P(S) = 1$ for sample space $S$; (3) for mutually exclusive events, $P(A_1 \cup A_2 \cup \cdots) = \sum_i P(A_i)$.

5. Define sample space and event in probability.

   The sample space $S$ is the set of all possible outcomes of a random experiment. An event is any subset of $S$.

6. What is the addition rule (inclusion-exclusion) for two events $A$ and $B$?

   $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

7. What is the probability of the complement of event $A$?

   $$P(A^{c}) = 1 - P(A)$$

8. When are two events $A$ and $B$ independent?

   When the occurrence of one does not affect the other, characterized by $$P(A \cap B) = P(A)\,P(B)$$

9. When are two events mutually exclusive (disjoint)?

   When they cannot occur together, i.e. $A \cap B = \varnothing$, so $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$.

10. Can two events with nonzero probability be both mutually exclusive and independent?

    No. If they are mutually exclusive, $P(A\cap B)=0$, but independence requires $P(A\cap B)=P(A)P(B)>0$. These contradict, so they cannot be both.

11. Define conditional probability of $A$ given $B$.

    $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$$

12. What is the multiplication (joint probability) rule for two events?

    $$P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)$$

13. What is marginal probability, and how is it obtained from a joint distribution?

    Marginal probability is the probability of a single event regardless of others, obtained by summing (or integrating) the joint probability over the other variable: $P(X=x) = \sum_y P(X=x, Y=y)$.

14. State Bayes' Theorem for events $A$ and $B$.

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

15. State the law of total probability for a partition $\{A_1, \ldots, A_n\}$ of the sample space.

    $$P(B) = \sum_{i=1}^{n} P(B \mid A_i)\,P(A_i)$$

16. What is the law of total expectation (tower property)?

    $$E[X] = E\big[E[X \mid Y]\big]$$

17. State the law of total variance.

    $$\operatorname{Var}(X) = E\big[\operatorname{Var}(X \mid Y)\big] + \operatorname{Var}\big(E[X \mid Y]\big)$$

18. Define the conditional expectation $E[X \mid Y=y]$ for discrete random variables.

    $$E[X \mid Y=y] = \sum_x x\, P(X=x \mid Y=y)$$

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