Link Between Discrete and Continuous Distributions
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. For example, let us consider a binary discrete random variable taking −1 or 1 for values, with probability ½ each.
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:
where x1, …, xn are the discrete values accessible to the variable and p1, …, pn are the probabilities associated with these values.
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.
Read more about this topic: Probability Density Function
Famous quotes containing the words link between, link, discrete and/or continuous:
“The secret of biography resides in finding the link between talent and achievement. A biography seems irrelevant if it doesn’t discover the overlap between what the individual did and the life that made this possible. Without discovering that, you have shapeless happenings and gossip.”
—Leon Edel (b. 1907)
“This is what we fear—no sight, no sound,
No touch or taste or smell, nothing to think with,
Nothing to love or link with,
The anaesthetic from which none come round.”
—Philip Larkin (1922–1986)
“One can describe a landscape in many different words and sentences, but one would not normally cut up a picture of a landscape and rearrange it in different patterns in order to describe it in different ways. Because a photograph is not composed of discrete units strung out in a linear row of meaningful pieces, we do not understand it by looking at one element after another in a set sequence. The photograph is understood in one act of seeing; it is perceived in a gestalt.”
—Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)
“We read poetry because the poets, like ourselves, have been haunted by the inescapable tyranny of time and death; have suffered the pain of loss, and the more wearing, continuous pain of frustration and failure; and have had moods of unlooked-for release and peace. They have known and watched in themselves and others.”
—Elizabeth Drew (1887–1965)