Normal probability distributions part I



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  • The following topics will be covered in this lecture:
    • Uniform distribution
    • Normal distributions
    • Finding the area corresponding to a probability
    • Finding a score corresponding to an area
    • Critical values
    • General normal distributions

Continuous random variables

Histogram of probability distribution for two coin flips with x number of heads.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

  • So far our examples have focused on discrete random variables, e.g.:
    • Results of coin flips – \( x \) is modeled with a binomial distribution.
    • Results of success / failure trials – \( x \) is modeled with a binomial distribution.
    • Number of occurances in an interval – \( x \) is modeled with a Poisson distribution.
  • We will now turn our attention to continuous random variables, but we will use what we learned about discrete variables to motivate this.
  • Recall that the probability histogram had the property, \[ \begin{align} \text{Area of Rectangle }x_\alpha &= P(x=x_\alpha) \times 1\\ &= P(x=x_\alpha). \end{align} \]
  • We also saw that we have the property \[ \sum_{x_\alpha \in \mathbf{R}} P(x=x_\alpha) =1. \]
  • Putting the above two properties together, we know, \[ \sum_{x_\alpha \in \mathbf{R}} \text{Area of Rectangle }x_\alpha =1. \]
  • For continuous random variables, we in fact have the same property with a minor modification:
    Let \( f(x) \) describe a curve for a probability distribution. Then the total area under the curve \( f(x) \) equals \( 1 \), and the probability of any event \( A \) equals the associated area under \( f(x) \) for all \( x_\alpha \) in the case of \( A \).

Uniform distribution

Random variables are the numerical measure of the outcome of a random process.

Courtesy of Ania Panorska CC

  • A basic example of the area property is with the uniform distribution
    • Let’s suppose that we are studying some procedure where all outcomes are equally likely.
      • A very simple example is if you are asked to guess a random number between \( 1 \) and \( 10 \), but including decimals.
    • That is, we will suppose that guessing \( 1.23453453 \) is equally likely as guessing \( 5 \).
  • Viewed in the above,
    • Our random experiment is guessing some number.
    • The outcome is one guess.
    • The random variable \( x \) is assigned the value of the guess.
  • Because we allow arbitrary decimal expansions, there are infinitely many choices.
  • However, all choices lie in the finite range \( [1,10] \) and are equally likely.
  • Discuss with a neighbor: if the area under the curve \( f(x) \) for \( x_\alpha \) in \( [1, 10] \) must equal one, and the height of \( f(x) \) is constant, what is the height?

Uniform distribution continued