IIT Madras BS Probability and Probability Distributions

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Certainly! Below is a clear, step-by-step explanation of the IIT Madras BS Probability and Probability Distributions PDF course, with examples, questions, and detailed solutions for each major topic¹. The layout is designed for ease of understanding and covers all essential steps.

1. Data, Statistics, and Probability

Key Concepts

• Experiment: Any process or phenomenon studied statistically (e.g., tossing a coin, rolling a die).
• Outcome: Result of an experiment (e.g., “Heads”, “1”).
• Sample Space: Set of all possible outcomes (e.g., for a die: $ S = {1,2,3,4,5,6} $)¹.

Example

• Tossing a Coin: $ S = {Heads, Tails} $
• Rolling a Die: $ S = {1,2,3,4,5,6} $

Solved Example: Sample Space

Q: An urn contains 3 red, 3 blue, and 3 white marbles. If you draw one marble, what is the sample space?

A: $ S = {red, blue, white} $

2. Events and Operations

• Event: Subset of the sample space (e.g., “even number” for a die).
• Union ($ A \cup B $): Outcomes in either A or B.
• Intersection ($ A \cap B $): Outcomes in both A and B.
• Complement ($ A^c $): Outcomes not in A.
• Disjoint Events: Events with no common outcomes.

Solved Example: Union and Intersection

Q: For rolling a die, define $ A = $ even numbers, $ B = $ prime numbers. Find $ A \cup B $ and $ A \cap B $.

A:

• $ A = {2,4,6} $
• $ B = {2,3,5} $
• $ A \cup B = {2,3,4,5,6} $
• $ A \cap B = {2} $

3. Probability Basics

• Probability Function ($ P $): Assigns to each event a number between 0 and 1.
• Axioms:
  • $ P(S) = 1 $
  • For disjoint events $ E_1, E_2, ··· $, $ P(E_1 \cup E_2 \cup ···) = P(E_1) + P(E_2) + ··· $

Solved Example: Probability Function

Q: For a fair coin, define $ P(Heads) = 0.5 $, $ P(Tails) = 0.5 $. Does this satisfy axioms?

A: Yes, because $ P(S) = P(Heads) + P(Tails) = 1 $.

4. Probability Distributions

• Uniform Distribution: All outcomes equally likely.
• Probability of an event: $ P(A) = \frac{Number of outcomes in A}{Total outcomes} $

Solved Example: Uniform Distribution

Q: An urn has 5 red and 8 blue marbles. What is the probability of drawing a red marble?

A:

• Total marbles = 13
• Red marbles = 5
• $ P(red) = \frac{5}{13} $

5. Conditional Probability

• Conditional Probability: $ P(A \mid B) = \frac{P(A \cap B)}{P(B)} $
• Multiplication Rule: $ P(A \cap B) = P(B) \cdot P(A \mid B) $

Solved Example: Conditional Probability

Q: A family has 2 children. At least one is a girl. What is the probability both are girls?

A:

• Sample space: $ {BB, BG, GB, GG} $
• Given at least one girl: $ {BG, GB, GG} $
• Both girls: $ {GG} $
• $ P(both girls \mid at least one girl) = \frac{1/4}{3/4} = \frac{1}{3} $

6. Bayes’ Theorem and Independence

• Bayes’ Theorem: $ P(B \mid A) = \frac{P(A \mid B) P(B)}{P(A)} $
• Independence: $ P(A \cap B) = P(A) P(B) $

Solved Example: Bayes’ Theorem

Q: 1% of people have Swine Flu. Test is 95% accurate for those with Swine Flu and 2% false positive. A person tests positive. What is the probability they have Swine Flu?

A:

• $ P(Disease) = 0.01 $
• $ P(Positive \mid Disease) = 0.95 $
• $ P(Positive \mid No Disease) = 0.02 $
• $ P(Positive) = 0.01 \times 0.95 + 0.99 \times 0.02 = 0.0293 $
• $ P(Disease \mid Positive) = \frac{0.01 \times 0.95}{0.0293} \approx 0.3242 $

7. Repeated Independent Trials

Bernoulli Distribution

• Single Trial: Success (1) with probability $ p $, failure (0) with $ 1-p $.
• Repeated Trials: $ n $ independent Bernoulli trials.

Solved Example: Bernoulli Trials

Q: A fair coin is tossed 5 times. What is the probability of exactly 2 tails?

A:

• $ n = 5 $, $ p = 0.5 $
• $ P(2 tails) = \binom{5}{2} (0.5)^2 (0.5)^3 = 10 \times \frac{1}{32} = \frac{10}{32} $

8. Binomial Distribution

• Binomial PMF: $ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $
• Example: Number of heads in $ n $ coin tosses.

Solved Example: Binomial Distribution

Q: A biased coin ($ P(Head) = 0.3 $) is tossed 10 times. What is the probability of exactly 3 heads?

A:

• $ P(X = 3) = \binom{10}{3} (0.3)^3 (0.7)^7 \approx 0.2668 $

9. Geometric Distribution

• Geometric PMF: $ P(X = k) = (1-p)^{k-1} p $
• Example: Number of trials until first success.

Solved Example: Geometric Distribution

Q: In Ludo, roll a die until you get a 1. What is the probability you need fewer than 6 throws?

A:

• $ p = \frac{1}{6} $
• $ P(X < 6) = 1 - (1-p)^5 = 1 - \left(\frac{5}{6}\right)^5 \approx 0.5981 $

10. Discrete Random Variables

• Random Variable: Function mapping outcomes to real numbers.
• PMF: $ f_X(t) = P(X = t) $

Solved Example: PMF

Q: A fair coin is tossed 3 times. Let $ X $ be the number of heads. Find the PMF.

A:

• $ X $ can be 0, 1, 2, or 3.
• $ P(X=0) = \frac{1}{8} $
• $ P(X=1) = \frac{3}{8} $
• $ P(X=2) = \frac{3}{8} $
• $ P(X=3) = \frac{1}{8} $

11. Common Discrete Distributions

12. Functions of Random Variables

• Function of RV: $ Y = g(X) $
• PMF of $ Y $: $ f_Y(a) = \sum_{t: g(t)=a} f_X(t) $

Solved Example: Function of RV

Q: $ X \sim Uniform{-2, -1, 0, 1, 2} $. Let $ g(X) = X^2 $. Find the PMF of $ g(X) $.

A:

• $ g(X) $ can be 0, 1, or 4.
• $ f_{g(X)}(0) = P(X=0) = \frac{1}{5} $
• $ f_{g(X)}(1) = P(X=-1) + P(X=1) = \frac{2}{5} $
• $ f_{g(X)}(4) = P(X=-2) + P(X=2) = \frac{2}{5} $

Summary Table

This layout and step-by-step approach make the concepts and calculations clear and easy to follow, with all steps and reasoning included for each example and question¹.