Degenerate distribution

In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. If the degenerate distribution is univariate (involving only a single random variable) it is a deterministic distribution and takes only a single value. Examples include a two-headed coin and rolling a die whose sides all show the same number. This distribution satisfies the definition of 'random variable' even though it does not appear random in the everyday sense of the word; hence it is considered degenerate. In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. If the degenerate distribution is univariate (involving only a single random variable) it is a deterministic distribution and takes only a single value. Examples include a two-headed coin and rolling a die whose sides all show the same number. This distribution satisfies the definition of 'random variable' even though it does not appear random in the everyday sense of the word; hence it is considered degenerate. In the case of a real-valued random variable, the degenerate distribution is localized at a point k0 on the real line. The probability mass function equals 1 at this point and 0 elsewhere. The degenerate univariate distribution can be viewed as the limiting case of a continuous distribution whose variance goes to 0 causing the probability density function to be a delta function at k0, with infinite height there but area equal to 1. The cumulative distribution function of the univariate degenerate distribution is: F k 0 ( x ) = { 1 , if  x ≥ k 0 0 , if  x < k 0 {displaystyle F_{k_{0}}(x)=left{{egin{matrix}1,&{mbox{if }}xgeq k_{0}\0,&{mbox{if }}x<k_{0}end{matrix}} ight.} In probability theory, a constant random variable is a discrete random variable that takes a constant value, regardless of any event that occurs. This is technically different from an almost surely constant random variable, which may take other values, but only on events with probability zero. Constant and almost surely constant random variables, which have a degenerate distribution, provide a way to deal with constant values in a probabilistic framework. Let  X: Ω → R  be a random variable defined on a probability space  (Ω, P). Then  X  is an almost surely constant random variable if there exists k 0 ∈ R {displaystyle k_{0}in mathbb {R} } such that and is furthermore a constant random variable if Note that a constant random variable is almost surely constant, but not necessarily vice versa, since if  X  is almost surely constant then there may exist  γ ∈ Ω  such that  X(γ) ≠ k0  (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ k0) = 0).