Probability mass function

In probability theory and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.^[1] The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose 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.^[2]

Formal definition

Suppose that X: S → A (A $\subseteq$ R) is a discrete random variable defined on a sample space S. Then the probability mass function f_X: A → [0, 1] for X is defined as^[3]^[4]

f_{X}(x)=\Pr(X=x)=\Pr(\{s\in S:X(s)=x\}).

Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes x:

\sum _{x\in A}f_{X}(x)=1

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 of X. That is, f_X may be defined for all real numbers and f_X(x) = 0 for all x $\notin$ X(S) as shown in the figure.

Since the image of X is countable, the probability mass function f_X(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]}

Measure theoretic formulation

A probability mass function of a discrete random variable X can be seen as a special case of two more general measure theoretic constructions: the distribution of X and the probability density function of X with respect to the counting measure. We make this more precise below.

Suppose that $(A,{\mathcal {A}},P)$ is a probability space and that $(B,{\mathcal {B}})$ is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of B. In this setting, a random variable $X\colon A\to B$ is discrete provided its image is a countable set. The pushforward measure $X_{*}(P)$ ---called a distribution of X in this context---is a probability measure on B whose restriction to singleton sets induces a probability mass function $f_{X}\colon B\to \mathbb {R}$ since $f_{X}(b)=P(X^{-1}(b))=[X_{*}(P)](\{b\})$ for each b in B.

Now suppose that $(B,{\mathcal {B}},\mu )$ is a measure space equipped with the counting measure. The probability density function f' of X with respect to the counting measure, if it exists, is the Radon-Nikodym derivative of the pushforward measure of X (with respect to the counting measure), so $f=dX_{*}P/d\mu$ and f is a function from B to the non-negative reals. As a consequence, for any b in B we have

P(X=b)=P(X^{-1}(\{b\})):=\int _{X^{-1}(\{b\})}dP=\int _{\{b\}}fd\mu =f(b),

demonstrating that f is in fact a probability mass function.

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

f_{X}(x)={\begin{cases}{\frac {1}{2}},&x\in \{0,1\},\\0,&x\notin \{0,1\}.\end{cases}}

This is a special case of the binomial distribution, the Bernoulli distribution.

An example of a multivariate discrete distribution, and of its probability mass function, is provided by the multinomial distribution.

References

^ Stewart, William J. (2011). Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press. p. 105. ISBN 978-1-4008-3281-1.
^ Probability Function at Mathworld
^ Kumar, Dinesh (2006). Reliability & Six Sigma. Birkhäuser. p. 22. ISBN 978-0-387-30255-3.
^ Rao, S.S. (1996). Engineering optimization: theory and practice. John Wiley & Sons. p. 717. ISBN 978-0-471-55034-1.

يک همريختي، نگاشتي است که ساختار انتخابي ميان دو ساختار جبري را حفظ ميکند. تعاريف ويژه براي همريختي شامل موارد زير است:  يک همريختي نيمگروهي، يک نگاشت است که عمليات باينري شرکتپذيري را حفظ ميکند.  يک همريختي نيمگروه يکهدار، يک همريختي نيمگروهي است که عنصر هماني را بر هماني همدامنه تصوير مي-کند.  يک همريختي گروهي، يک همريختي است که ساختار گروه را حفظ ميکند. اين همريخت ممکن است به طور همارز به عنوان همريختي نيمگروهي بين گروهها تعريف گردد.  يک همريختي حلقهاي، يک همريختي است که ساختار حلقهاي را حفظ ميکند. که کدام عضو هماني ضرب بر مبناي تعريف حلقهي مورد استفاده بايد حفظ گردد.  يک نگاشت خطي، همريختي است که ساختار فضاي برداري را حفظ ميکند، يعني ساختار گروه جابجايي و ضرب اسکالر. نوع اسکالر بايد بيشتر به منظور تعيين کردن همريختي نشان داده شود به عنوان مثال هر نگاشت R-linear يک نگاشت Z-linear است اما عکس آن صحيح نميباشد.  يک همريختي مدول، يک نگاشت است که ساختارهاي مدول را حفظ ميکند.  يک همريختي جبرها، يک همريختي است که ساختار جبري را حفظ ميکند.  يک تابعگر، يک همريخت بين دو گروه است. لزومي ندارد که تمام ساختاري که متعلق به يک شيء است، توسط يک همريخت حفظ گردد. به عنوان مثال، ممکن است يک ساختار داراي همريختي نيمگروهي بين دو نيمگروه يکهدار باشد. در صورتيکه اين ساختار، هماني دامنهي تعريف را برهماني همدامنه تصوير نکند، يک همريختي نيمگروهي يکهدار نخواهد بود. به عنوان مثال، يک گروه يک شي جبري متشکل از يک مجموعه همراه با يک عمل دوتايي منفرد است که از بديهيات اوليهي معيني پيروي ميکند. اگر (G, *) و (H, *' ) گروه باشند، همريختي از (G, *) به (H, *' ) تابع f ميباشد:f: (G, *)→(H, *' ) به طوريکه براي تمام عضوهاي G ∈ g1, g2 داريم: f(g1 ∗ g2)= f(g1) ∗′ f(g2). از آنجا که Gو H داراي معکوس ميباشند، ميتوان نشان داد که با تصوير کردن هماني G بر هماني H ، معکوس آنها تغيير نميکند. ساختار جبري حفظ شده ممکن است شامل بيش از يک عمل باشد و لازم باشد همريختي هر عمل را حفظ کند. به عنوان مثال، يک حلقه داراي جمع و ضرب است و هم-ريختي از حلقه (R, +, ∗, 0, 1)به حلقه (R′, +′, ∗′, 0′, 1′) تابعي نظير f(r + s) = f(r) +′ f(s), f(r ∗ s) = f(r) ∗′ f(s) است که براي هر عضو r و s از حلقهي دامنه f(1) = 1′ . در صورتيکه حلقهها تاکيد بر يکهدار بودن نداشته باشند، شرط آخر حذف ميگردد. همچنين، اگر ساختارهاي تعريف شده ( به عنوان مثال، معکوسهاي 0 و جمعي در مورد يک حلقه) لزوما توسط موارد بالا حفظ نگردد، نگاهداشتن آنها الزاماتي را خواهد افزود.

مفهوم همريختي ميتواند يک تعريف صوري در زمينهي جبر جهاني که ايدههاي مشترک در تمام ساختارهاي جبري را مطالعه ميکند، ارائه دهد.

[1] Stewart, William J. (2011). Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press. p. 105. ISBN 978-1-4008-3281-1.

[2] Probability Function at Mathworld

[3] Kumar, Dinesh (2006). Reliability & Six Sigma. Birkhäuser. p. 22. ISBN 978-0-387-30255-3.

[4] Rao, S.S. (1996). Engineering optimization: theory and practice. John Wiley & Sons. p. 717. ISBN 978-0-471-55034-1.

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v t e Theory of probability distributions
probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function
raw moment central moment mean variance standard deviation skewness kurtosis L-moment
moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant

Formal definition

Measure theoretic formulation

Examples

References

Further reading