In-Depth Notes on Normal and Uniform Probability Distributions

Properties of the Normal Distribution

  • Definition: The normal distribution is a continuous probability distribution characterized by a bell-shaped curve, symmetric about the mean.
  • Key Characteristics:
    • Mean (μ): The center of the distribution, where half the values are on either side.
    • Standard Deviation (σ): Measures the spread of the distribution. About 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three (known as the empirical rule).
  • The standard normal distribution has a mean of 0 and a standard deviation of 1.

Uniform Probability Distribution

  • Example: Delivery time from UPS, represented as a random variable X ranging from 0 to 60 minutes (10 am to 11 am).
  • Characteristics:
    • All intervals of equal length within the range [0, 60] are equally likely.
    • Probability Density Function (pdf): Must satisfy two properties:
    1. Total area under the curve equals 1.
    2. Height of the curve ≥ 0.
  • Graph Example: The height of the rectangle is calculated as 1/60 for the uniform distribution.
    • Area under the curve represents the probability of random variable falling within a certain interval.

Applications of the Normal Distribution

  • Standardizing a Normal Random Variable:

    • Any random variable X that is normally distributed can be transformed into a standard normal variable Z using the formula: Z = (X - μ) / σ.
    • This allows for easier calculations involving probabilities.

  • Finding Area Under the Normal Curve:

    • Using statistical software or calculators, specific areas (probabilities) can be computed using the cumulative distribution function (normalcdf).

  • Complement Rule: For finding areas on the right side of the curve, use:

    • Area to the right of z = 1 = 1 - Area to the left of z = 1.

Examples of Finding Probabilities

  • Using z-scores: When given the specifics, you apply a standard normal distribution table or function to find probabilities or values.
    • Example: Given a score of 450 seconds with σ = 60, find P(time < 312 seconds).
    • Calculation: Use normalcdf function after standardizing the appropriate values.

Conclusion - Z-scores and Area Calculations

  • Z-scores: Important for determining the position of a value within the distribution.
    • For a specific area, find zα such that the area to the right is known.
  • Understanding Curve Behavior: Normal curves approach the horizontal axis but never touch it.
  • Mathematical Functions: You use normalcdf and invNorm in calculators to compute specific probabilities and find specific x-values under the normal distribution.