 Entropy of a Bernoulli trial as a function of success probability, called the binary entropy function. In information theory, the binary entropy function, denoted H(p) or Hb(p), is defined as the entropy of a Bernoulli trial with probability of success p. Mathematically, the Bernoulli trial is modelled as a random variable X that can take on only two values: 0 and 1. The event X = 1 is considered a success and the event X = 0 is considered a failure. (These two events are mutually exclusive and exhaustive.) Information entropy of a Bernoulli trial X. If X can assume values 0 and 1, entropy of X is defined as H(X) = -Pr(X=0) log2 Pr(X=0) - Pr(X=1) log2 Pr(X=1). ...
Information entropy of a Bernoulli trial X. If X can assume values 0 and 1, entropy of X is defined as H(X) = -Pr(X=0) log2 Pr(X=0) - Pr(X=1) log2 Pr(X=1). ...
In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ...
A bundle of optical fiber. ...
Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function. ...
In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ...
This article does not cite its references or sources. ...
A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...
If Pr(X = 1) = p, then Pr(X = 0) = 1 − p and the entropy of X is given by
 The logarithms in this formula are usually taken (as shown in the graph) to the base 2. See binary logarithm. Plot of log2 x In mathematics, the binary logarithm (log2 n) is the logarithm for base 2. ...
When  the binary entropy function attains its maximum value. This is the case of the unbiased bit, the most common unit of information entropy. A bit (binary digit) refers to a digit in the binary numeral system, which consists of base 2 digits (ie. ...
Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function. ...
H(p) is distinguished from the entropy function by its taking a single scalar constant parameter. For tutorial purposes, in which the reader may not distinguish the appropriate function by its argument, H2(p) is often used; however, this could confuse this function with the analogous function related to Rényi entropy, so Hb(p) (with "b" not in italics) should be used to dispel ambiguity. Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function. ...
In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ...
In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...
The factual accuracy of this article is disputed. ...
In information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. ...
Derivative
The derivative of the binary entropy function may be expressed as the negative of the logit function: In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...
In mathematics, especially as applied in statistics, the logit (pronounced with a long o and a soft g, IPA ) of a number p between 0 and 1 is Plot of logit in the range 0 to 1, base is e (The base of the logarithm function used here is of...
See also A bundle of optical fiber. ...
Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function. ...
References - David J. C. MacKay. Information Theory, Inference, and Learning Algorithms Cambridge: Cambridge University Press, 2003. ISBN 0-521-64298-1
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