Review__Chapters_3__4__5_

CIS 2033 Review Part 2

• Instructor: Dr. Abha Belorkar

• Course: CIS 2033, Temple University

• Term: Fall 2020

Chapter 3

Topics Covered

• Poisson Distribution

• Hypergeometric Distribution

• Expected Value and Variance

Example Problem

• Problem Statement: Given 15 CIS and 10 Math faculty members, calculate the probability of selecting exactly 3 CIS faculty when choosing 7 members at random.

Poisson Distribution

Introduction

• Poisson Random Variable: Models the count of occurrences of events within a given unit of time, distance, area, or volume.

Conditions for Poisson Distribution

1. Events must occur independently (e.g., emails arriving independently).

2. The theoretical rate of occurrence is constant (e.g., average number of houses sold in a month).

Application Examples

• Fatal commercial airline crashes in a year can be modeled by Poisson.

• The number of deaths in commercial airline crashes cannot be modeled by Poisson due to dependency on the number of crashes.

Example Problem

• Email Frequency: On weekdays, the average is 1/30 emails per minute; on weekends, it's 1 email per minute. 1. Probability Calculation: What is the probability of receiving no emails over a 4-hour interval on a Sunday?

  • Let X: number of emails in 4 hours on Sunday, where the mean is 8 emails.

  • Poisson Probability Formula: P(X=0) = e^(-λ) * (λ^x/x!)

  • Here, λ = 8; thus, P(X=0) = e^(-8).

Chapter 4

Topics Covered

• Integration

• Calculating CDF (Cumulative Distribution Function)

• Uniform Distribution

• Normal Distribution

• Exponential Distribution

Example Problems

1. Finding Probabilities: Given f(x) = x^2 / 3 for -1 < x < 2. Calculate:

   • a) P(0 < X < 1)

   • b) CDF of X, denoted as F_X(x).

   • c) Value of F_X(1).

2. Uniform Distribution Problem: Arrival times uniformly distributed between 1 and 2 PM.

   • a) P(T=30)?

   • b) P(T ∈ [0,30))?

   • c) For 0 < a < b < 60, find P(a < T < b).

3. Battery Life Problem: A battery lasts on average 3.0 years with a standard deviation of 0.5 year. Calculate the probability of lasting less than 2.3 years.

   • Use Z-score computation: Z = (X - μ) / σ.

Chapter 5

Topics Covered

• Double Integration

• Continuous case

• Discrete case