probability density function

Examples of probability density function in the following topics:

• 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)$.

• The Density Scale

  • Density estimation is the construction of an estimate based on observed data of an unobservable, underlying probability density function.
  • Density estimation is the construction of an estimate based on observed data of an unobservable, underlying probability density function.
  • 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.

• Continuous Sampling Distributions

  • Moreover, in continuous distributions, the probability of obtaining any single value is zero.
  • Therefore, these values are called probability densities rather than probabilities.
  • 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 probability for the random variable to fall within a particular region is given by the integral of this variable's density over the region .
  • Boxplot and probability density function of a normal distribution $N(0, 2)$.

• Summary of the Uniform and Exponential Probability

  • Probability density function: f (X) = $\frac{1}{ ba }$ for a≤ X ≤b

• From histograms to continuous distributions

  • This smooth curve represents a probability density function (also called a density or distribution), and such a curve is shown in Figure 2.28 overlaid on a histogram of the sample.
  • A density has a special property: the total area under the density's curve is 1.
  • Density for heights in the US adult population with the area between 180 and 185 cm shaded.

• 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.
  • The resulting probability distribution of the random variable can be described by a probability density, where the probability is found by taking the area under the curve.
  • 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.

• 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 general, calculus is needed to find the area under the curve for many probability density functions.

• 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.
  • We define the function f (x) so that the area between it and the x-axis is equal to a probability.
  • 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.

• Practice 1: Uniform Distribution

  • Exercise 5.6.8: Find the probability that a randomly chosen car in the lot was less than 4 years old.
  • Exercise 5.6.9: Out of just the cars less than 7.5 years old, find the probability that a randomly chosen car in the lot was less than 4 years old.

• Multiple Regression Models

  • Multiple regression is used to find an equation that best predicts the $Y$ variable as a linear function of the multiple $X$ variables.
  • The purpose of a multiple regression is to find an equation that best predicts the $Y$ variable as a linear function of the $X$ variables.
  • Multiple regression would give you an equation that would relate the tiger beetle density to a function of all the other variables.
  • For example, if you did a regression of tiger beetle density on sand particle size by itself, you would probably see a significant relationship.
  • If you did a regression of tiger beetle density on wave exposure by itself, you would probably see a significant relationship.