4. Probability and Statistics

  • Classical probability model and methods for probability calculation.

  • Numerical characteristics of data:

  • Mean

  • Variance

  • Basic concepts pertaining to normal distribution.

Classical Probability Model

  • The classical probability model assumes that all outcomes are equally likely.

  • Probability is calculated as: P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)} where

    • $P(E)$ is the probability of event E occurring,

    • $n(E)$ is the number of favorable outcomes,

    • $n(S)$ is the total number of outcomes in the sample space.

Methods for Probability Calculation

  1. Empirical Probability

    • Based on observed data rather than a theoretical model.

    • Calculated as:
      P(E)=n(E)n(total)P(E) = \frac{n(E)}{n(total)}

  2. Theoretical Probability

    • Based on the reasoning behind probability.

    • Uses the classical model for equally likely outcomes.

  3. Subjective Probability

    • Based on personal judgment or experience rather than exact calculation.

Numerical Characteristics of Data

  1. Mean

    • The average of a set of values.

    • Calculated as:
      Mean=∑x<em>in\text{Mean} = \frac{\sum{x<em>i}}{n} where $xi$ are the individual values and $n$ is the number of values.

  2. Variance

    • A measure of how much the values deviate from the mean.

    • Calculated as:
      Variance=∑(xi−Mean)2n\text{Variance} = \frac{\sum{(x_i - \text{Mean})^2}}{n}

    • A higher variance indicates greater spread in the data.

Normal Distribution

  • A continuous probability distribution that is symmetric about the mean.

  • Characterized by its bell-shaped curve.

  • Key properties:

    • The mean, median, and mode are all equal.

    • Approximately 68% of the data falls within one standard deviation of the mean.

    • Approximately 95% falls within two standard deviations.

    • Approximately 99.7% falls within three standard deviations.

  • The standard normal distribution has a mean of 0 and a standard deviation of 1, denoted as Z-distribution:
    Z=(X−μ)σZ = \frac{(X - \mu)}{\sigma}
    where (X) is a value, (\mu) is the mean, and (\sigma) is the standard deviation.