Intro starts week 2 pt1

Class Announcements

• Emphasize that students can participate by raising their hand.

• Reinforce that the classroom is a safe space for questions; passing is acceptable without penalty.

• Highlight recent changes regarding lab vacancies: students may attend earlier labs if free, to avoid long waiting times.

Introduction to Probability

• Will cover probabilities again briefly before focusing on discrete probability distributions: especially Poisson and binomial distributions.

Definition of Probability

• Probability refers to the chance of an event occurring; it ranges between 0 (impossible) and 1 (certain).

• Example: Probability of rain = 0 (impossible) vs. probability of a sunny day = 1 (certain).

• Probability can be expressed as both decimals and percentages; acceptable formats for exam answers.

Basic Probability Examples

• Coin Toss:

  • Probability of heads = 1/2 or 0.5.

  • Probability of tails = 1/2 or 0.5.

• Dice Roll:

  • Probability of rolling a four on a six-sided die = 1/6.

  • Better to keep probability as fractions for precision.

Notation in Probability

• Random Variables:

  • Denoted by X, Y, etc.

  • Example: P(X=x) indicates the probability of the random variable X equating to x.

  • For a six-sided die, P(X=5) = 1/6.

• Card example: Probability of Y (ace of spades) = 4/52 = 1/13.

Discrete Probability Distribution

Characteristics

• Definition: Lists the probability of each outcome in an experiment (only integer values).

• Example of a probability distribution for a six-sided die:

  • Outcomes: 1, 2, 3, 4, 5, 6 with each has P=1/6.

• Tasks regarding given probability distribution tables during tests (calculating probabilities).

Properties of Probability Distributions

• Probability values must always range between 0 and 1.

• The sum of probabilities in a distribution must equal to 1.

Validating a Probability Distribution

• Check sums and ranges:

  • Valid if:

    • All values are between 0 and 1.

    • Sum of all probabilities equals 1.

  • Common pitfalls include negative values and exceeding the sum of one.

Expected Value Calculation (Mean)

• Denoted by μ (mu) or E(X).

• Calculation formula:

  • E(X) = Σ (x_i * P(x_i)), summing across all outcomes.

• Example demonstrated with a series of outcomes (0, 1, 2) to compute expected value with given probabilities.

Standard Deviation in Probability Distributions

• Denoted by σ (sigma).

• Calculation requires:

  • Each outcome squared multiplied by its probability, summed, then square root of the variance.

  • The detailed steps for calculating standard deviation demonstrated.

Specific Discrete Probability Distributions

Poisson Distribution

• Models number of occurrences in a fixed interval of time or space.

  • Events must occur independently at a known constant mean (μ).

• Important terms defined:

  • X = number of occurrences.

  • μ = average rate of occurrence.

• Example: The number of calls to a call center in one hour.

• Notation: X ~ Poisson(μ).

• Mean = μ, Standard Deviation = √μ.

Binomial Distribution

• Formula and application not yet explored in depth, but will address in further discussions.

Probability Tables

• Two main types:

  • Individual (exact probabilities).

  • Cumulative (probabilities up to a certain point).

• Strategies for using these tables:

  • Find appropriate row for given mean/parameter.

  • For cumulative, sum values from zero to the limit given in problem.

Problem Solving with Probability Distributions

• Use case examples to clarify concepts and test applications through real-world scenarios.

• Example question involving probability of at most a certain number of events, using cumulative tables.

• Emphasis on using complementary probability for finding probabilities that exceed a certain number.

Using R for Probability Calculations

• Practical examples demonstrated using R programming:

  • Employing dpois() and ppois() functions for discrete and cumulative probabilities, respectively.