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.

• Determining Sample Size

  • As follows, this can be estimated by pre-determined tables for certain values, by Mead's resource equation, or, more generally, by the cumulative distribution function.
  • Let $X_i, i = 1, 2, \dots, n$, be independent observations taken from a normal distribution with unknown mean $\mu$ and known variance $\sigma^2$.
  • If $z_{\alpha}$ is the upper $\alpha$ percentage point of the standard normal distribution, then:
  • Calculate the appropriate sample size required to yield a certain power for a hypothesis test by using predetermined tables, Mead's resource equation or the cumulative distribution function.

• 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

• Single-Population Inferences

  • One standard choice for an approximating distribution is the empirical distribution of the observed data.
  • $K$-sample Anderson–Darling tests are available for testing whether several collections of observations can be modeled as coming from a single population, where the distribution function does not have to be specified.
  • It makes use of the fact that, when given a hypothesized underlying distribution and assuming the data does arise from this distribution, the data can be transformed to a uniform distribution.
  • The formula for the test statistic $A$ to assess if data $\{ Y_1 < \dots, n \}$ comes from a distribution with cumulative distribution function (CDF) $F$ is:
  • Note that in this case no parameters are estimated in relation to the distribution function $F$.

• 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.

• 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 $-1$.

• 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.