University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

74

Pages:

3
Author:

Niklas

Definition of Probability Mass Function: Given a random variable X with finite support {x1, x2, ..., xn}, the probability mass function of X is defined as: P(X = xi) = p(xi) Example: Flipping a Coin 3 Times RV of Heads on 3 tosses: 0H: 1/8 1H: 3/8 2H: 3/8 3H: 1/8 P(H) = 50% P(T) = 50% Example: Number of Tails in Tossing a Coin 3 Times Random variable X can assume values {0, 1, 2, 3} Example: X is a discrete random variable X ∈ {0, 1, 2, 3, 4} n = 4 p = 0.4 Probability function: P(X = xi) = (n choose xi) * p^xi * (1-p)^(n-xi) Table and Graph: A table and graph are provided to illustrate the probability mass function and cumulative distribution function for the given example. Introduction: Introduces the concept of expectation as a measure of the center of a probability distribution. Expectation of a Discrete Random Variable: Defines the expected value E(X) of a discrete random variable X as the weighted average of the possible values of X. Provides the formula for calculating E(X). Example: Presents an example of a random variable X representing the number of errors on a page. Calculates the probability mass function for X. Computes the expected value of X. Variance of a Discrete Random Variable: Defines the variance V(X) of a discrete random variable X as the expectation of the squared deviations from the mean. Provides the formula for calculating V(X). Introduces the standard deviation as the positive square root of the variance. Example: Continues the previous example and calculates the variance and standard deviation of X. Binomial Distribution: States the expected value and variance for a binomial random variable with parameters n and p. Poisson Distribution: Describes the Poisson distribution as a probability distribution for the number of independent occurrences in a given interval of time. Lists examples of random phenomena that can be modeled using the Poisson distribution. Definition of Poisson Distribution: A random variable X follows a Poisson distribution with parameter λ if and only if: P(X = x) = (λ^x / x!) * e^(-λ) X~Poisson(λ) represents the number of occurrences/successes of a certain event in a given continuous interval. Assumptions: The interval is divided into a large number of equal subintervals. The probability of occurrence in each subinterval is small. The probability of occurrence is constant for all subintervals. Occurrences are independent. Properties of Poisson Distribution: For a Poisson random variable X with parameter λ: E(X) = λ V(X) = λ Example: Football Goals: A football team scores an average of 1.5 goals per game. The number of goals per game is assumed to follow a Poisson distribution. Calculate the probability of scoring 0, 1, 2, 3, 4, or more goals in the next game. Calculations: The probability of scoring x goals is calculated using the Poisson probability mass function with λ = 1.5. The probabilities for different values of x are summed to find the probability of scoring more than 4 goals.