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[[Image:Discrete probability distrib.svg|right|thumb|The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.]]
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In [[probability theory]] and [[statistics]], a '''probability mass function''' ('''pmf''') is a function that gives the probability that a [[discrete random variable|discrete]] [[random variable]] is exactly equal to some value.<ref>{{cite book|author=Stewart, William J.|title=Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling|publisher=Princeton University Press|year=2011|isbn=978-1-4008-3281-1|page=105|url=http://books.google.com/books?id=ZfRyBS1WbAQC&pg=PT105}}</ref> The probability mass function is often the primary means of defining a [[discrete probability distribution]], and such functions exist for either [[Scalar variable|scalar]] or [[multivariate random variable]]s whose [[Domain of a function|domain]] is discrete.
 
A probability mass function differs from a [[probability density function]] (pdf) in that the latter is associated with continuous rather than discrete random variables; the values of the latter are not probabilities as such: a pdf must be integrated over an interval to yield a probability.<ref>[http://mathworld.wolfram.com/ProbabilityFunction.html Probability Function] at Mathworld </ref>
 
==Formal definition==
[[Image:Fair dice probability distribution.svg|right|thumb|The probability mass function of a [[Dice|fair die]]. All the numbers on the {{dice}} have an equal chance of appearing on top when the die stops rolling.]]
 
Suppose that ''X'': ''S'' → ''A'' (A <math>\subseteq</math> '''R''') is a [[discrete random variable]] defined on a [[sample space]] ''S''. Then the probability mass function ''f''<sub>''X''</sub>: ''A'' → [0, 1] for ''X'' is defined as<ref>{{cite book|author=Kumar, Dinesh|title=Reliability & Six Sigma|publisher=Birkhäuser|year=2006|isbn=978-0-387-30255-3|page=22|url=http://books.google.com/books?id=XsX20uCFJbYC&pg=PA22}}</ref><ref>{{cite book|author=Rao, S.S.|title=Engineering optimization: theory and practice|publisher=John Wiley & Sons|year=1996|isbn=978-0-471-55034-1|page=717|url=http://books.google.com/books?id=nuoryE4IwMoC&pg=PA717}}</ref>
:<math>f_X(x) = \Pr(X = x) = \Pr(\{s \in S: X(s) = x\}).</math>
 
Thinking of probability as mass helps avoiding mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes ''x'':
:<math>\sum_{x\in A} f_X(x) = 1</math>
 
When there is a natural order among the hypotheses ''x'', it may be convenient to assign numerical values to them (or ''n''-tuples in case of a discrete [[multivariate random variable]]) and to consider also values not in the [[Image_(mathematics)|image]] of ''X''. That is, ''f''<sub>''X''</sub> may be defined for all [[real number]]s and ''f''<sub>''X''</sub>(''x'') = 0 for all ''x'' <math>\notin</math> ''X''(''S'') as shown in the figure.
 
Since the image of ''X'' is [[countable]], the probability mass function ''f''<sub>''X''</sub>(''x'') is zero for all but a countable number of values of ''x''. The discontinuity of probability mass functions is related to the fact that the [[cumulative distribution function]] of a discrete random variable is also discontinuous. Where it is differentiable, the derivative is zero, just as the probability mass function is zero at all such points.{{Citation needed|date=April 2012}}
 
==Examples==
Suppose that ''S'' is the sample space of all outcomes of a single toss of a fair coin, and ''X'' is the random variable defined on ''S'' assigning 0 to "tails" and 1 to "heads". Since the coin is fair, the probability mass function is
:<math>f_X(x) = \begin{cases}\frac{1}{2}, &x \in \{0, 1\},\\0, &x \notin \{0, 1\}.\end{cases}</math>
This is a special case of the [[binomial distribution]].
 
An example of a multivariate discrete distribution, and of its probability mass function, is provided by the [[multinomial distribution]].
 
==References==
{{refimprove|date=April 2012}}
{{reflist}}
 
==Further reading==
*Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9 (p&nbsp;36)
 
{{Theory of probability distributions}}
 
[[Category:Probability theory]]
[[Category:Types of probability distributions]]

Latest revision as of 04:07, 19 December 2014

You have been bombarded lately by a lot of ads online or in TV trying to recommend the only diets which function for we. Yet, none of these guaranteed weight reduction plans did not make any outcome or you have lost couple pounds, yet not for lengthy.

Whatever choice of diet we choose to take, the advice of the doctor or nutritionist is important for your success plus healthy fat reduction. If you follow the above mentioned regulations then the experience is enjoyable plus fruitful. You should have realistic goals set for the weight reduction to avoid disillusionment while paying close attention to ensure we completely understand what are the best diet plans quick for females the planet over.

It refuses to matter how little the fat reduction is. When it comes to diet plans, should you find 1 that offers you a result then you've got a winner. remain with it!

Whole grains: if you need to take grainy foods, go for beans and/ brown rice. Foods that are wealthy inside complete grains mean which its outer kernel, vitamin plus inner mineral haven't been removed.

Popular fad diets by definition are usually several type of deprivation to the body. Usually calories, the average amount a day on many of the diets is anywhere from 800 to 1600 calories. As we undertake the diet and your body has less food to task, it will take the opportunity to receive rid of excess toxins in a program. This takes many water which is why many of the weight loss found on the diet is water fat. As soon as we go back to the standard eating practices, the body may replace the water it lost plus you'll gain the weight back.

A perfect approach to do healthy weight loss success is to join all types of diet food into healthy eating plans, taking benefit of the advantages of each ingredient. Capsaicin will be combined with caffeine, green tea, plus cinnamon to provide a moderate and sensible increase inside calories burned every day. The role of capsaicin is to stimulate the sympathetic nervous system. This component offers only the positive plus healthy effects.

To conclude, no matter where we are with a weight initial thing you need to do is love the body now. It helps you a great deal of wonderful aspects, so not body bash. With all love you give it, being healthy takes small effort.

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