- Probability mass function
In

probability theory , a**probability mass function**(abbreviated**pmf**) is a function that gives the probability that a discreterandom variable is exactly equal to some value. A pmf differs from aprobability density function (abbreviated**pdf**) in that the values of a pdf, defined only forcontinuous random variable s, are not probabilities as such. Instead, the integral of a pdf over a range of possible values ("a", "b"] gives the probability of the random variable falling within that range. See notation for the meaning of ("a", "b"] .**Mathematical description**Suppose that "X" is a discrete random variable, taking values on some

countable sample space "S" ⊆**R**. Then the probability mass function "f"_{"X"}("x") for "X" is given by:$f\_X(x)\; =\; egin\{cases\}\; Pr(X\; =\; x),\; xin\; S,\backslash 0,\; xin\; mathbb\{R\}ackslash\; S.end\{cases\}$Note that this explicitly defines "f"_{"X"}("x") for allreal number s, including all values in**R**that "X" could never take; indeed, it assigns such values a probability of zero.The discontinuity of probability mass functions reflects the fact that the

cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable (i.e. where "x" ∈**R**"S") the derivative is zero, just as the probability mass function is zero at all such points.**Example**Suppose that "X" is the outcome of a single coin toss, assigning 0 to tails and 1 to heads. The probability that "X" = "x" is 0.5 on the state space {0, 1} (this is a Bernoulli random variable), and hence the probability mass function is:$f\_X(x)\; =\; egin\{cases\}frac\{1\}\{2\},\; x\; in\; \{0,\; 1\},\backslash 0,\; x\; in\; mathbb\{R\}ackslash\{0,\; 1\}.end\{cases\}$

**ee also***

Discrete probability distribution **References**Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9 (p36)

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