# Random variable and density function

Probability distributions of random variables Video transcript In the last video, I introduced you to the notion of-- well, really we started with the random variable. And then we moved on to the two types of random variables. You had discrete, that took on a finite number of values. And the these, I was going to say that they tend to be integers, but they don't always have to be integers. ## Statistics Glossary - random variables and probability distributions

Invertible functions In the case in which the function is neither strictly increasing nor strictly decreasing, the formulae given in the previous sections for discrete and absolutely continuous random variables are still applicable, provided is one-to-one and hence invertible.

We report these formulae below. One-to-one functions of a discrete random variable When is a discrete random variable, the probability mass function of is given by the following.

Systems Simulation: The Shortest Route to Applications. This site features information about discrete event system modeling and simulation. It includes discussions on descriptive simulation modeling, programming commands, techniques for sensitivity estimation, optimization and goal-seeking by simulation, and what-if analysis. Here, we will define jointly continuous random variables. Basically, two random variables are jointly continuous if they have a joint probability density function as defined below. A probability density function captures the probability of being close to a number even when the probability of any single number is zero.

Proposition probability mass of a one-to-one function Let be a discrete random variable with support and probability mass function be one-to-one on the support of.

Then, the support of and its probability mass function is Proof The proof of this proposition is identical to the proof of the propositions for strictly increasing and strictly decreasing functions found above: One-to-one functions of an absolutely continuous random variable When is an absolutely continuous random variable and is differentiable, then also is absolutely continuous and its probability density function is given by the following proposition.

Proposition density of a one-to-one function Let be an absolutely continuous random variable with support and probability density function be one-to-one and differentiable on the support of.

Then, the support of.Glossary of Statistical Terms You can use the "find" (find in frame, find in page) function in your browser to search the glossary. Here, we will define jointly continuous random variables. Basically, two random variables are jointly continuous if they have a joint probability density function as defined below. To determine the distribution of a discrete random variable we can either provide its PMF or CDF. For continuous random variables, the CDF is well-defined so we can provide the CDF.

In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.

The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent. In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

As a function, a random variable is required to be measurable, which rules out certain pathological cases where the quantity which the random variable returns is infinitely sensitive to small changes in the outcome.

Heuristically, the probability density function is just the distribution from which a continuous random variable is drawn, like the normal distribution, which is the .

Distribution Function -- from Wolfram MathWorld