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In probability theory , a probability density function PDF , or density of a continuous random variable , is a function whose value at any given sample or point in the sample space the set of possible values taken by the random variable can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values , as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1. The terms " probability distribution function "  and " probability function "  have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function , or it may be a probability mass function PMF rather than the density.
A random variable is a numerical description of the outcome of a statistical experiment. A random variable that may assume only a finite number or an infinite sequence of values is said to be discrete; one that may assume any value in some interval on the real number line is said to be continuous. For instance, a random variable representing the number of automobiles sold at a particular dealership on one day would be discrete, while a random variable representing the weight of a person in kilograms or pounds would be continuous. The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable. For a discrete random variable, x , the probability distribution is defined by a probability mass function, denoted by f x.
Instead, we can usually define the probability density function PDF. The PDF is the density of probability rather than the probability mass. The concept is very similar to mass density in physics: its unit is probability per unit length. Nevertheless, as we will discuss later on, this is not important. Figure 4. The uniform distribution is the simplest continuous random variable you can imagine.
Sign in. In my first and second introductory posts I covered notation, fundamental laws of probability and axioms. These are the things that get mathematicians excited. However, probability theory is often useful in practice when we use probability distributions. Probability distributions are used in many fields but rarely do we explain what they are. Often it is assumed that the reader already knows I assume this more than I should. For example, a random variable could be the outcome of the roll of a die or the flip of a coin.
These ideas are unified in the concept of a random variable which is a numerical summary of random outcomes. Random variables can be discrete or continuous. A basic function to draw random samples from a specified set of elements is the function sample , see? We can use it to simulate the random outcome of a dice roll. The cumulative probability distribution function gives the probability that the random variable is less than or equal to a particular value.
Sign in. In my first and second introductory posts I covered notation, fundamental laws of probability and axioms. These are the things that get mathematicians excited. However, probability theory is often useful in practice when we use probability distributions. Probability distributions are used in many fields but rarely do we explain what they are.
A continuous random variable takes on an uncountably infinite number of possible values. We'll do that using a probability density function "p. We'll first motivate a p. Even though a fast-food chain might advertise a hamburger as weighing a quarter-pound, you can well imagine that it is not exactly 0. One randomly selected hamburger might weigh 0.
Analyze properties, find moments and determine the likelihood of outcomes of continuous distributions. Compute properties of a continuous distribution:. Analyze properties, find moments and determine the likelihood of outcomes of discrete distributions. Compute properties of a discrete distribution:.
Note that mgf is an alternate definition of probability distribution. Hence there is one for one relationship between the pdf and mgf. However mgf does not exist.