probability distribution function

ASU W.P. Carey School of Business

  • Arizona State University

  • Course: Data Distribution

  • Instructor: Dr. Asish Satpathy

Learning Goals

  1. Distribution of continuous variables

  2. Distribution of discrete variables

  3. Random Variables

  4. Probability Distribution of Random Variables

Description for Continuous Variables

  • Continuous Data Variables: Used to describe data that can take any value within a given range.

    • Centrality Metrics:

      • Mean

      • Median

      • Mode

    • Spread Metrics:

      • Quartiles

      • Range

      • Standard Deviation

      • Variance

      • Interquartile Range

      • Mean Absolute Deviation

    • Shape Indicators:

      • Skewness

      • Kurtosis

Distribution of Continuous Variable

  • Mean: -0.246661

  • Standard Deviation: 1.0010272

  • Number of Observations (N): 50

  • Variance: 1.0020554

  • Skewness: -0.154412

  • Kurtosis: -0.345945

  • Range: 4.345398

Description for Discrete Variables

  • Discrete Data Variables: Represent data that can only take certain distinct values.

    • Centrality Metrics:

      • Mode (derived from frequency graphs, but not particularly meaningful)

    • Shape: Defined by the distribution of frequencies.

Distribution of Discrete Variables

  • Frequencies:

    • Impressionism: Count - 13, Probability - 0.23636

    • Landscape: Count - 13, Probability - 0.23636

    • Modern: Count - 15, Probability - 0.27273

    • Performance: Count - 8, Probability - 0.14545

    • Renaissance: Count - 6, Probability - 0.10909

    • Total Count: 55, Total Probability: 1.00000

Random Variables

  • Continuous Random Variable Examples:

    • Amount of rain in Tempe in the next month.

    • Length of time waiting for a table at a restaurant.

  • Discrete Random Variable Examples:

    • Number of bounced checks on a given day at the bank.

    • Number of drinks ordered by first customers at a Starbucks.

Probability Distribution for Continuous Random Variables

  • Describes the probability associated with continuous variables.

  • Graph insights:

    • Represents chance (not finite data).

    • Probability for a customer to wait between A and B minutes is the area under the curve.

    • Waiting time probability from 0 to 60 minutes equals 1 (total area under curve).

Probability Distribution for Discrete Random Variables

  • Visualized using a probability histogram.

  • Example for number of heads after three flips of a fair coin:

    • Outcomes and probabilities are as follows:

      • 0 Heads: Probability = 1/8

      • 1 Head: Probability = 3/8

      • 2 Heads: Probability = 3/8

      • 3 Heads: Probability = 1/8

Normal and t-Distribution Functions

  • Graphical Representation of normal distribution function.

    • Depicts the shape and probabilities related to a standard normal distribution.

    • t-distribution is used when n is close to or smaller than 30.

Poisson Distribution (Non-normal)

  • Describes the probability of a given number of events occurring in a fixed interval of time or space.

  • Related probabilities depicted on a graphical function.

Test Your Knowledge

  • Question 1: Which of the following measures is not used in describing a continuous data variable?

    • A: Mean

    • B: Variance

    • C: Frequency

    • D: Kurtosis

  • Question 2: Which of the following is an example of a discrete random variable?

    • A: Number of children in a family

    • B: The amount of sugar in an orange

    • C: Time required to run a five-mile marathon

    • D: Profit margin in the next quarter