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:
“This sand seemed to us the connecting link between land and water. It was a kind of water on which you could walk, and you could see the ripple-marks on its surface, produced by the winds, precisely like those at the bottom of a brook or lake. We had read that Mussulmans are permitted by the Koran to perform their ablutions in sand when they cannot get water, a necessary indulgence in Arabia, and we now understand the propriety of this provision.”
—Henry David Thoreau (1817–1862)
“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)
“We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the child’s life, these words are discrete and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”
—Selma H. Fraiberg (20th century)
“I read the newspapers avidly. It is my one form of continuous fiction.”
—Aneurin Bevan (1897–1960)