cumulative distribution function

Examples of cumulative distribution function in the following topics:

• Continuous Probability Distributions

  • A continuous probability distribution is a probability distribution that has a probability density function.
  • Mathematicians also call such a distribution "absolutely continuous," since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure $\lambda$.
  • The definition states that a continuous probability distribution must possess a density; or equivalently, its cumulative distribution function be absolutely continuous.
  • This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions—singular distributions, which are neither continuous nor discrete nor a mixture of those.
  • The standard normal distribution has probability density function:

• Properties of Continuous Probability Distributions

  • The curve is called the probability density function (abbreviated: pdf).
  • We use the symbol f (x) to represent the curve. f (x) is the function that corresponds to the graph; we use the density function f (x) to draw the graph of the probability distribution.
  • Area under the curve is given by a different function called the cumulative distribution function (abbreviated: cdf).
  • The cumulative distribution function is used to evaluate probability as area.
  • In this chapter and the next chapter, we will study the uniform distribution, the exponential distribution, and the normal distribution.

• Optional Collaborative Classrom Activity

  • The probability density function is a rather complicated function.
  • The cumulative distribution function is P ( X < x ) .
  • As the notation indicates, the normal distribution depends only on the mean and the standard deviation.
  • This means there are an infinite number of normal probability distributions.
  • One of special interest is called the standard normal distribution.

• Two Types of Random Variables

  • The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution.
  • Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability.
  • Continuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function (CDF) that is absolutely continuous.
  • The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution.
  • This shows the probability mass function of a discrete probability distribution.

• The Uniform Distribution

  • The distribution is often abbreviated $U(a, b)$.
  • The uniform distribution is useful for sampling from arbitrary distributions.
  • A general method is the inverse transform sampling method, which uses the cumulative distribution function (CDF) of the target random variable.
  • The probability density function is written as:
  • Contrast sampling from a uniform distribution and from an arbitrary distribution

• Cumulative Frequency Distributions

  • A cumulative frequency distribution displays a running total of all the preceding frequencies in a frequency distribution.
  • A cumulative frequency distribution is the sum of the class and all classes below it in a frequency distribution.
  • Rather than displaying the frequencies from each class, a cumulative frequency distribution displays a running total of all the preceding frequencies.
  • Constructing a cumulative frequency distribution is not that much different than constructing a regular frequency distribution.
  • There are a number of ways in which cumulative frequency distributions can be displayed graphically.

• Income Distribution

  • Income distribution can be described as a functional or personal distribution.
  • The functional distribution of income describes the allocation of income among the factors of production.
  • The distribution of income among the members of society, individuals and families, is called the personal distribution of income.
  • A Lorenz curve plots the cumulative proportion of income units and cumulative proportion of income received when income units are arrayed from lowest to highest.
  • A Lorenz curve is usually shown as plots of the cumulative proportion of income units and cumulative proportion of income received when income units are arrayed from lowest to highest.

• The Normal Distribution

  • Normal distributions are a family of distributions all having the same general shape.
  • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
  • The normal distribution is the only absolutely continuous distribution whose cumulants, other than the mean and variance, are all zero.
  • This is written as N (0, 1), and is described by this probability density function:
  • This function is symmetric around $x=0$, where it attains its maximum value $\frac { 1 }{ \sqrt { 2\pi } }$; and has inflection points at $+1$ and $-1$.

• Types of Innovation

  • The second is a technology-based functional mode in which the development of new technology drives innovation.
  • These challenges can include developing new products and services or new administrative techniques and technologies for performing organizational functions (e.g., production, marketing, and sales and distribution).
  • Walmart's "Hub and Spoke" distribution model is a classic example of strategic innovation.
  • Cumulative innovation is any instance of something new being created from more than one source.
  • Remixing music is a direct example of cumulative innovation.

• Relative Frequency Distributions

  • Constructing a relative frequency distribution is not that much different than from constructing a regular frequency distribution.
  • Create the frequency distribution table, as you would normally.
  • Just like we use cumulative frequency distributions when discussing simple frequency distributions, we often use cumulative frequency distributions when dealing with relative frequency as well.
  • Cumulative relative frequency (also called an ogive) is the accumulation of the previous relative frequencies.
  • To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.