Probability Distributions and Sampling Distributions Notes

  • Probability Distributions

  • Probability Density Function (pdf)

    • Describes likelihood of continuous outcomes for random variables
    • Must sum to 1
    • Calculable for events
    • Notation: f(x)

  • Probability Mass Function (pmf)

    • Describes likelihoods for discrete random variables
    • Masses sum to one
    • Notation: p(x)

  • Cumulative Probability Functions

  • Cumulative Distribution Function (cdf)

    • Describes probability of all outcomes ≤ specific outcome
    • Shown as F(x) or P(X ≤ x)
    • Increasing in nature
    • Used for probabilities less than, greater than, or between values

  • Binomial Distribution

  • Models two mutually exclusive outcomes (e.g., success vs. failure)

  • Key parameters:

    • n = number of trials
    • p = probability of success

  • Trials assumed independent with constant p

  • Applications: quality control, customer satisfaction, loan defaults

  • Example: Coin toss to find the probability of getting 3 heads in 10 trials:

    • P(X=3) = (10 choose 3) * (0.5)^3 * (0.5)^7 = 0.117188

  • Example of Customer Shipment Acceptance

  • A shipment is acceptable if ≤10% defects.

  • If sampling 10 products (n=10, p=0.1), probability of rejection (≥2 defects) is 26.4%

  • Reducing the defect rate to 5% lowers rejection probability to 8.6%

  • Normal Distributions

  • Used to model continuous variables

  • Defined by mean (μ) and standard deviation (σ)

  • Symmetrical shape with total area = 1

  • Examples: Demand management with normal distribution of units sold

  • Key applications include calculating probabilities of being below/above specific thresholds

  • Excel Tools for Normal Distribution

  • NORM.DIST function used to find cumulative probability

    • P(X ≤ x) or P(X < x) calculations

  • NORM.INV function for finding specific demand levels corresponding to a probability

  • Examples of demand calculations based on provided μ and σ values

  • Standard Normal Distribution

  • Z-distribution with μ = 0 and σ = 1

  • Appears on the z-axis indicating standard deviations from mean

  • Z-scores facilitate comparison among different distributions

  • Central Limit Theorem (CLT)

  • As sample size increases (n > 30), sample means approach a normal distribution regardless of parent distribution

  • Assumptions include independence, identical distribution, and finite mean/variance

  • Sampling Distributions

  • Enables analysis of a sample to infer properties about a population

  • Importance of random sampling to minimize bias and ensure representativeness

  • Waiting Times in Customer Service

  • Average wait times affect customer satisfaction; expectations around 5-10 minutes

  • Statistical tools can help understand probability distributions of wait times

  • Calculating Waiting Times

  • Example: Average service time = 3.6 minutes; SD = 1 minute.

    • Probability (x > 5) = 1 - NORM.DIST(5, 3.6, 1, TRUE) = 0.0808
    • 8% chance of waiting over 5 minutes