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Can someone explain the word "admit" and explain what would happen if it does not admit a distribution?

Project Backlog
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    Can you use it in a sentence? I'm not sure I've heard the word 'admit' in this context before. –  Oct 02 '14 at 05:38
  • From Wiki - "If a random variable admits a probability density function, then the characteristic ... functions of distributions defined by the weighted sums of random variables" https://www.google.ca/search?q=random+variable+admits+distribution&rlz=1C1CHWA_enCA601CA601&oq=random+variable+admits+distribution&aqs=chrome..69i57.6578j0j7&sourceid=chrome&es_sm=93&ie=UTF-8 – Project Backlog Oct 02 '14 at 05:51
  • admit and induce are often used interchangebly. "admits a fixed point" "induces a fixed point", etc. – Alan Oct 02 '14 at 05:54

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Not every probability distribution on the real numbers can be described by a probability density function: for example, the probability distribution given by (on the variable $X$)

$$ P(X = a) = \begin{cases} 1/2 & a=0,1 \\ 0 & \text{otherwise} \end{cases} $$

cannot be expressed by a probability density function.

"Admit" is sometimes used to express the notion that you're allowed or able to do something -- in this case, saying that a distribution "admits a density function" means that the distribution the phrase refers to really does have a probability density function.

  • I thought any Probability Mass Function could be transformed into a PDF through the use of Dirac Delta spikes, in this case we would have two spikes of height 1/2 at a = 0 and 1 – Project Backlog Oct 02 '14 at 06:02
  • @Project: Dirac deltas aren't functions! One can certainly make use of an appropriate generalization of the notion of function, though. –  Oct 02 '14 at 06:11
  • "Not every probability distribution on the real numbers" - what do you mean by "probability distribution" here? My textbook uses it interchangeable with pdf, yet the paper I'm reading says "If the real data distribution P_r admits a density..." Thanks! – Andriy Drozdyuk Dec 17 '18 at 18:02
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A random variable, $X$, is a measurable function from a probability space $(\Omega,\mathcal{F},\mathbb{P})$ to $(\mathbb{R},\mathcal{B}_{\mathbb{R}})$. The induced measure on $\mathcal{B}_{\mathbb{R}}$, defined by $$\mu(A)=\mathbb{P}(X^{-1}(A))$$ is the distribution of $X$.

QED
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