Chapter4 A (1)

Discrete Random Variables

Chapter Overview

• Focus on Discrete Random Variables in STA205 (Spring 2021).

Definition

• Discrete Random Variable: A variable that can take on a countable number of possible values.

Probability Distribution and Histograms

Probability Distribution

• Similar to a relative frequency table.

• Lists all possible values and their corresponding probabilities.

Probability Histogram

• Similar to a relative frequency histogram.

• Values are on the horizontal axis, probabilities are on the vertical axis.

Difference between Relative Frequency and Probability

Example Scenario

• A class of 20 students with the following distribution:

  • Relative Frequency Table Example:

    • of Sisters | Frequency | Relative Frequency

    • 0 | 4 | 4/20 = 0.2 (20%)

    • 1 | 10 |

    • 2 | 6 |

Key Concepts

• Relative frequency shows the proportion of observations in each category.

• Probability gives the chance of a random variable assuming a certain value.

Probability Calculations

Random Variable X

• Let X denote the number of sisters of a randomly selected student.

  • P(X=0) = 0.2

  • P(X=1) = 0.5

  • P(X=2) = 0.3

• Probability Distribution of X:

  • Sum of probabilities must equal 1: ΣP(X=x) = 1.

At Least and At Most Events

• P(X ≤ x): Probability that X is less than or equal to x.

  • Example: P(X has at most 1 sister) = P(X=0) + P(X=1) = 0.7

• P(X ≥ x): Probability that X is greater than or equal to x.

  • Example: P(X has at least 1 sister) = 0.8.

Example Problem: Family with Children

Define Random Variable X

• X = Number of girls in a family with 4 children.

• Possible values: 0, 1, 2, 3, 4.

Probability Distribution Outcomes

• 16 equally likely possibilities.

  • Outcomes include all permutations of boys and girls (e.g., BBBB, BBBG, etc.).

• Probability Distribution Example:

  • of Girls | Probability

  • 0 | 1/16

  • 1 | 4/16

  • 2 | 6/16

  • 3 | 4/16

  • 4 | 1/16

Probability of Various Outcomes

• P(X=2) = 6/16 (Probability of exactly 2 girls).

• P(X ≥ 2) = 11/16 (Probability of at least two girls).

• P(X ≤ 2) = 11/16 (Probability of at most 2 girls).

Example Problem: Probabilities from Given Criteria

Problem Breakdown

• 1: If P(X < 2) = 0.2, find P(X=2).

  • Calculation: P(X < 2) = P(X=0) + P(X=1) = 0.2.

  • P(X ≥ 3) = 0.7 leads to P(X=2) = 0.1.

• 2: If P(X ≤ 3) = 0.4, and P(X > 2) = 0.9, find P(X=3).

Mean and Standard Deviation

Mean of a Discrete Random Variable

• Mean (expected value) μ defined as: μ = ΣxP(X=x).

• Example Calculation:

  • Hourly Pay Example:

    • Workers' hourly pay: [4, 4, 5, 5, 5, 6, 6, 6, 6, 8].

    • Mean = ($5.5/hr).

Standard Deviation

• Standard deviation (σ) defined as: σ = Σ(x-μ)²P(X=x).

• Example Calculation for Workers' Hourly Pay:

  • σ calculated as √1.25 = 1.118.

Expected Value Calculation Example

• GPA: 3.7 with probability 0.9, and GPA: 3.5 with probability 0.1.

• Expected GPA = 3.70.9 + 3.50.1 = 3.68.