GATE Statistics Probability Syllabus

Probability syllabus — full chapter and topic list

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

1. 1

   Axiomatic definition of probability

   3 topics

   • Properties of probability function
   • Conditional probability
   • Bayes' theorem
2. 2

   Random variables and their distributions

   10 topics

   • Distribution function
   • Probability mass function
   • Probability density function and their properties
   • Expectation
   • Moments and moment generating function
   • Quantiles
   • Distribution of functions of a random variable
   • Chebyshev inequality
   • Markov inequality
   • Jensen inequality

Probability flashcards for GATE Statistics

22 of 51 cards from the Probability deck — real questions with worked answers.

1. State the three axioms (Kolmogorov axioms) of a probability function $P$ defined on events of a sample space $\Omega$.

   1. Non-negativity: $P(A) \geq 0$ for every event $A$. 2. Normalization: $P(\Omega) = 1$. 3. Countable additivity: for pairwise disjoint events $A_1, A_2, \dots$, $P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i)$.

2. What is the complement rule and the probability of the empty set for a probability function?

   $P(A^{c}) = 1 - P(A)$ and $P(\varnothing) = 0$.

3. State the monotonicity property of probability.

   If $A \subseteq B$, then $P(A) \leq P(B)$, and moreover $P(B \setminus A) = P(B) - P(A)$.

4. State the general addition rule (inclusion–exclusion) for two events $A$ and $B$.

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

5. Write the inclusion–exclusion formula for three events $A$, $B$, $C$.

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

6. State Boole's inequality (the union bound) for events $A_1,\dots,A_n$.

   $$P\left(\bigcup_{i=1}^{n} A_i\right) \leq \sum_{i=1}^{n} P(A_i).$$

7. Define the conditional probability of $A$ given $B$, and state the condition required.

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

8. State the multiplication (chain) rule for the joint probability of events $A_1, A_2, \dots, A_n$.

   $$P(A_1\cap\cdots\cap A_n) = P(A_1)\,P(A_2\mid A_1)\,P(A_3\mid A_1\cap A_2)\cdots P(A_n\mid A_1\cap\cdots\cap A_{n-1}).$$

9. When are two events $A$ and $B$ independent, and what does this imply for conditional probability?

   $A$ and $B$ are independent iff $P(A\cap B) = P(A)\,P(B)$. Then $P(A\mid B) = P(A)$ (when $P(B)>0$).

10. State the law of total probability for a partition $\{B_1,\dots,B_n\}$ of the sample space.

    $$P(A) = \sum_{i=1}^{n} P(A\mid B_i)\,P(B_i),$$ where the $B_i$ are mutually exclusive and exhaustive with $P(B_i)>0$.

11. State Bayes' theorem for a partition $\{B_1,\dots,B_n\}$ and event $A$.

    $$P(B_k\mid A) = \frac{P(A\mid B_k)\,P(B_k)}{\sum_{i=1}^{n} P(A\mid B_i)\,P(B_i)}.$$

12. In Bayes' theorem, name the prior, the likelihood, and the posterior.

    Prior: $P(B_k)$ (belief before observing data). Likelihood: $P(A\mid B_k)$ (probability of the evidence under hypothesis $B_k$). Posterior: $P(B_k\mid A)$ (updated belief after observing $A$).

13. Define the (cumulative) distribution function $F_X$ of a random variable $X$.

    $$F_X(x) = P(X \leq x), \quad x \in \mathbb{R}.$$

14. List the defining properties of a cumulative distribution function $F$.

    1. Non-decreasing. 2. Right-continuous: $\lim_{h\to 0^{+}} F(x+h) = F(x)$. 3. $\lim_{x\to -\infty} F(x) = 0$ and $\lim_{x\to +\infty} F(x) = 1$.

15. How do you compute $P(a < X \leq b)$ from the CDF $F_X$?

    $$P(a < X \leq b) = F_X(b) - F_X(a).$$

16. Using a CDF $F_X$, express $P(X = a)$ in terms of left and right limits.

    $$P(X = a) = F_X(a) - F_X(a^{-}),$$ the size of the jump at $a$; it equals $0$ wherever $F_X$ is continuous.

17. Define the probability mass function (pmf) of a discrete random variable and give its two defining conditions.

    $p_X(x) = P(X = x)$. Conditions: $p_X(x) \geq 0$ for all $x$, and $\sum_{x} p_X(x) = 1$ over the support.

18. Define the probability density function (pdf) of a continuous random variable $X$ in relation to its CDF.

    $f_X(x) = \dfrac{d}{dx} F_X(x)$ wherever the derivative exists, equivalently $F_X(x) = \int_{-\infty}^{x} f_X(t)\,dt$.

19. List the defining properties of a probability density function $f_X$.

    1. $f_X(x) \geq 0$ for all $x$. 2. $\int_{-\infty}^{\infty} f_X(x)\,dx = 1$. 3. $P(a \leq X \leq b) = \int_{a}^{b} f_X(x)\,dx$.

20. For a continuous random variable, what is $P(X = a)$ and why?

    $P(X = a) = 0$ for any single point $a$, because $\int_{a}^{a} f_X(x)\,dx = 0$. Hence $P(a\leq X\leq b) = P(a < X < b)$.

21. Define the expectation $E[X]$ for discrete and for continuous random variables.

    Discrete: $E[X] = \sum_{x} x\,p_X(x)$. Continuous: $E[X] = \int_{-\infty}^{\infty} x\,f_X(x)\,dx$, provided the sum/integral converges absolutely.

22. State the law of the unconscious statistician (LOTUS) for $E[g(X)]$.

    Discrete: $E[g(X)] = \sum_{x} g(x)\,p_X(x)$. Continuous: $E[g(X)] = \int_{-\infty}^{\infty} g(x)\,f_X(x)\,dx$.

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