The probability mass function of a discrete random variable is the density with respect to the counting measure over the sample space usually the set of integers , or some subset thereof. It is not possible to define a density with reference to an arbitrary measure e. Furthermore, when it does exist, the density is almost everywhere unique. The standard normal distribution has probability density. If a random variable X is given and its distribution admits a probability density function f , then the expected value of X if the expected value exists can be calculated as.

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution , even though it has no discrete component, i. A distribution has a density function if and only if its cumulative distribution function F x is absolutely continuous. In this case: F is almost everywhere differentiable , and its derivative can be used as probability density:.

Two probability densities f and g represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero. In the field of statistical physics , a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function. This alternate definition is the following:. It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function, by using the Dirac delta function.

The density of probability associated with this variable is:. More generally, if a discrete variable can take n different values among real numbers, then the associated probability density function is:. This substantially unifies the treatment of discrete and continuous probability distributions. For instance, the above expression allows for determining statistical characteristics of such a discrete variable such as its mean , its variance and its kurtosis , starting from the formulas given for a continuous distribution of the probability. It is common for probability density functions and probability mass functions to be parametrized—that is, to be characterized by unspecified parameters.

It is important to keep in mind the difference between the domain of a family of densities and the parameters of the family.

Different values of the parameters describe different distributions of different random variables on the same sample space the same set of all possible values of the variable ; this sample space is the domain of the family of random variables that this family of distributions describes.

A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the normalization factor of a distribution the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1.

This normalization factor is outside the kernel of the distribution. Since the parameters are constants, reparametrizing a density in terms of different parameters, to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones. Changing the domain of a probability density, however, is trickier and requires more work: see the section below on change of variables.

For continuous random variables X 1 , This density function is defined as a function of the n variables, such that, for any domain D in the n -dimensional space of the values of the variables X 1 , This is called the marginal density function, and can be deduced from the probability density associated with the random variables X 1 , Continuous random variables X 1 , If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable.

This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. However, rather than computing. The values of the two integrals are the same in all cases in which both X and g X actually have probability density functions.

## Probability density function - Wikipedia

It is not necessary that g be a one-to-one function. In some cases the latter integral is computed much more easily than the former.

See Law of the unconscious statistician. This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,. The above formulas can be generalized to variables which we will again call y depending on more than one other variable.

## Probability concepts explained: probability distributions (introduction part 3)

Then, the resulting density function is [ citation needed ]. This derives from the following, perhaps more intuitive representation: Suppose x is an n -dimensional random variable with joint density f. This result leads to the Law of the unconscious statistician :. Applying the change of variable theorem from the previous section we obtain that. The probability density function of the sum of two independent random variables U and V , each of which has a probability density function, is the convolution of their separate density functions:.

It is possible to generalize the previous relation to a sum of N independent random variables, with densities U 1 , Then, the joint density p y , z can be computed by a change of variables from U,V to Y,Z , and Y can be derived by marginalizing out Z from the joint density.

### Concept of Random Variable

Or you may want to know the probability that a standard normal random variable lies between two given values. These probabilities are easy to compute from a normal distribution table. Here's how. In school or on the Advanced Placement Statistics Exam, you may be called upon to use or interpret standard normal distribution tables.

Standard normal tables are commonly found in appendices of most statistics texts. Often, phenomena in the real world follow a normal or near-normal distribution. This allows researchers to use the normal distribution as a model for assessing probabilities associated with real-world phenomena. Typically, the analysis involves two steps. The problem in the next section demonstrates the use of the normal distribution as a tool to model real-world events.

- Introduction;
- Characterizing a Distribution — Introduction to Statistics documentation.
- Variance of an Arbitrary function of a random variable g(X).
- Between Empire and Alliance: America and Europe during the Cold War.
- 11. Probability Distributions - Concepts!
- EXAMPLE: Foot Length!
- Covariance.

Molly earned a score of on a national achievement test. The mean test score was with a standard deviation of What proportion of students had a higher score than Molly? Assume that test scores are normally distributed.

The correct answer is B. As part of the solution to this problem, we assume that test scores are normally distributed.