monotonic function

Examples of monotonic function in the following topics:

• Comparing Two Populations: Independent Samples

  • It assesses how well the relationship between two variables can be described using a monotonic function.
  • If there are no repeated data values, a perfect Spearman correlation of $1$ or $-1$ occurs when each of the variables is a perfect monotone function of the other.

• Distribution-Free Tests

  • non-parametric statistics (in the sense of a statistic over data, which is defined to be a function on a sample that has no dependency on a parameter), whose interpretation does not depend on the population fitting any parameterized distributions.
  • Spearman's rank correlation coefficient: measures statistical dependence between two variables using a monotonic function.

• Rank Correlation

  • A Spearman correlation of 1 results when the two variables being compared are monotonically related, even if their relationship is not linear.

• Common Discrete Probability Distribution Functions

  • Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson.
  • A probability distribution function is a pattern.

• Continuous Probability Distributions

  • A continuous probability distribution is a probability distribution that has a probability density function.
  • In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value.
  • Unlike a probability, a probability density function can take on values greater than one.
  • The standard normal distribution has probability density function:
  • Boxplot and probability density function of a normal distribution $$$N(0, 2)$.

• Continuous Probability Functions

  • We begin by defining a continuous probability density function.
  • We use the function notation f (x).
  • In the study of probability, the functions we study are special.
  • Consider the function f (x) = 1 20 for 0 ≤ x ≤ 20. x = a real number.
  • This particular function, where we have restricted x so that the area between the function and the x-axis is 1, is an example of a continuous probability density function.

• The Density Scale

  • Density estimation is the construction of an estimate based on observed data of an unobservable, underlying probability density function.
  • The unobservable density function is thought of as the density according to which a large population is distributed.
  • A probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
  • The above image depicts a probability density function graph against a box plot.
  • This image shows a boxplot and probability density function of a normal distribution.

• Exercises

  • For the following data, plot the theoretically expected z score as a function of the actual z score (a Q-Q plot).
  • For the "SAT and College GPA" case study data, create a contour plot looking at College GPA as a function of Math SAT and High School GPA.

• Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function X ~ H (r,b,n)

  • NOTE : Currently, the TI-83+ and TI-84 do not have hypergeometric probability functions.

• Continuous Sampling Distributions

  • A probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
  • Boxplot and probability density function of a normal distribution $N(0, 2)$.