Study Notes on Discrete Random Variables and Their Probability Distributions

MATH 2303: Introduction to Probability & Statistics

Overview

• Instructor: Ms. Galloway, MA Mathematics Education

• Institution: University of the Incarnate WordⓇ

Lecture 8 Outline

• Last Class:

  • Review of Exam 1 scheduled to be discussed on Tuesday.

• Today's Focus:

  • Chapter 5: Discrete Random Variables and Their Probability Distributions

    • 5.1 – Random Variables

    • 5.2 – Probability Distribution of a Discrete Random Variable

    • 5.3 – Mean and Standard Deviation of a Discrete Random Variable

Chapter 5: Discrete Random Variables and Their Probability Distributions

5.1 Random Variables

Definitions

• Random Variables:

  • A random variable is defined as a variable whose value is determined by the outcome of a random experiment.

• Discrete Random Variable:

  • A random variable that assumes countable values.

• Continuous Random Variable:

  • A random variable that can assume any value contained in one or more intervals.

Examples of Discrete Random Variables

1. The number of cars sold at a dealership during a given month.

2. The number of houses on a certain block.

3. The number of fish caught on a fishing trip.

4. The number of complaints received at an airline’s office on a given day.

5. The number of customers visiting a bank during any given hour.

6. The number of heads obtained in three tosses of a coin.

Examples of Continuous Random Variables

1. The length of a room.

2. The time taken to commute from home to work.

3. The amount of milk in a gallon (not necessarily exact, may slightly exceed or be less than one gallon).

4. The weight of a letter.

5. The price of a house (often treated as a continuous random variable due to a large number of unique values).

5.2 Probability Distribution of a Discrete Random Variable

Definition

• The probability distribution of a discrete random variable lists all possible values that the random variable can assume along with their corresponding probabilities.

Example 5-1

• Scenario: Frequency and relative frequency distributions of the number of vehicles owned by families sampled from 2000 families.

Probability Distribution for Example 5-1

Table:

Characteristics of a Probability Distribution

1. For each value of x, it holds that $0 \leq P(x) \leq 1$.

2. The sum of the probabilities must equal 1: $Σ P(x) = 1$.

Example 5-2: Calculating Probabilities

• Using the probability distribution:

  • (a) The probability that a randomly selected family owns two vehicles:

    • $P(selected \ family \ owns \ 2 \ vehicles) = P(2) = 0.425$.

  • (b) The probability that a randomly selected family owns at least two vehicles:

    • $P(at \ least \ two \ vehicles) = P(2 \ or \ 3 \ or \ 4) = P(2) + P(3) + P(4) = 0.425 + 0.245 + 0.080 = 0.750$.

  • (c) The probability that a randomly selected family owns at most one vehicle:

    • $P(at \ most \ 1) = P(0 \ or \ 1) = P(0) + P(1) = 0.015 + 0.235 = 0.250$.

  • (d) The probability that a randomly selected family owns three or more vehicles:

    • $P(3 \ or \ 4) = P(3) + P(4) = 0.245 + 0.080 = 0.325$.

Example 5-3: Validating Probability Distributions

• Given tables of probabilities, you must determine:

  • (a) Invalid: Sum of probabilities is not equal to 1.

  • (b) Valid: Probabilities sum to 1.

  • (c) Invalid: Contains a negative probability.

Example 5-4: Probability Distribution for Machine Breakdowns

• Breakdown Probability Distribution of machine:

• Question: What is the probability of breakdowns of 0, 1, 2, etc?

• Solution: Detailed analysis similar to prior examples will follow.

5.3 Mean and Standard Deviation of a Discrete Random Variable

Mean

• The mean (expected value) of a discrete random variable $x$, denoted as $µ$, is calculated as:
  $µ = Σ x P(x)$.

Example 5-6: Calculation of Mean

• Given the breakdown probability data, calculate as follows:

  • $µ = Σx P(x) = 1.80$.

Standard Deviation

• The standard deviation of a discrete random variable $x$, denoted as $s$, is a measure of the spread of its probability distribution, computed as:
  $s = \sqrt{Σ (x - µ)^2 P(x)}$.

Example 5-7: Standard Deviation Calculation for Defective Parts

• Given a probability distribution of defective parts, compute:

• Example: $s ≈ 1.204.$

Example 5-8: Company Profit Probability Distribution

1. Define annual profits based on sales performance with respective probabilities:

   • High Sales: Profit of $4.5 million with probability 0.32

   • Medium Sales: Profit of $1.2 million with probability 0.51

   • Low Sales: Loss of $2.3 million with probability 0.17

2. Calculate mean and standard deviation:

   • Distribution of $x$ shows profitable and loss scenarios clearly.

   • Implications on business strategy considered based on computed values.

Review Notes

• Last Class:

  • Review of Exam 1 (scheduled for Tuesday).

• Today's Focus:

  • Chapter 5 covering:

    • Random Variables (5.1)

    • Probability Distribution of a Discrete Random Variable (5.2)

    • Mean and Standard Deviation of a Discrete Random Variable (5.3)