Discrete distribution

Definition taught in Random variables and distributions, part of Probability

Discrete distribution

Mass, density and distribution function. From Random variables and distributions.
Type Definition
Taught in Random variables and distributions
Part Probability
Worked examples 2
Questions using it 8

X is discrete when it takes values in a countable set S. Its probability mass function is f(x)=\mathbb{P}(X=x) for x\in S, and \mathbb{P}(X\in B)=\sum_{x\in B}f(x).

Examples

The sum of two dice

Question page: The sum of two dice

Two fair dice are thrown in order, and Z is their sum.

  1. Write Z as a function on the sample space.

  2. Give the probability mass function of Z.

  1. On the sample space \{1,\ldots,6\}^2, Z(x,y)=x+y.

  2. The value z\in\{2,\ldots,12\} occurs in 6-|z-7| of the 36 pairs, so \mathbb{P}(Z=z)=(6-|z-7|)/36.

The smaller face

Question page: The smaller face

Two fair dice are thrown and Z is the smaller of the two faces.

  1. Give the probability mass function of Z in a table.

  2. Calculate \mathbb{E} Z.

  1. There are 36 equally likely ordered pairs. The number of pairs whose minimum is z is 11,9,7,5,3,1 for z=1,\ldots,6: the pairs in which both faces are at least z, minus the pairs in which both are at least z+1.
z 1 2 3 4 5 6
\mathbb{P}(Z=z) 11/36 9/36 7/36 5/36 3/36 1/36
  1. \mathbb{E} Z=(1\cdot 11+2\cdot 9+3\cdot 7+4\cdot 5+5\cdot 3+6\cdot 1)/36=91/36.

Figures

Three discrete laws: the number of successes in 10 trials with success probability 0.7, the trial of the first success when each trial succeeds with probability 0.3, and a Poisson count with mean 10. From Random variables and distributions.

Used in

See also

Conditional distribution · Expectation · Expectation of a function · Joint probability mass function · Maximum likelihood · Random sample · Random variable · Cumulative distribution function

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