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The Geometric Probability Distribution(几何分布)
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A Geometric experiment possesses the following properties:
1. The experiment consists of a series of identical trials.
2. Each trial results in one of two outcomes: success, S, or failure, F.
3. The probability of success on a single trial is equal to some value p and remains the same from trial
1. The experiment consists of a series of identical trials.
2. Each trial results in one of two outcomes: success, S, or failure, F.
3. The probability of success on a single trial is equal to some value p and remains the same from trial
to trial. The probability of a failure is equal to q = (1 − p).
4. The trials are independent.
5. The random variable of interest is Y , the number of failures before a success occurs.
4. The trials are independent.
5. The random variable of interest is Y , the number of failures before a success occurs.
Definition:
A random variable Y is said to have a geometric probability distribution if and only if
$$p(y) = {\left( {1 - p} \right)^{y}}p,\;\;\;\;y = 0,\;1,\;...,\;0 \le p \le 1.$$
You can use the following Mathematica command to obtain the probability
PDF[GeometricDistribution[p], y]
Relative Mathematica Functions
GeometricDistribution[p] represents a geometric distribution with success probability p.
Examples:
A = GeometricDistribution[p];
a := {Arrowheads[0.02], Arrow[{{4, PDF[A, 0]}, {0, PDF[A, 0]}}]};
t := Text[Style["p=" <> ToString[p], Medium], {4, PDF[A, 0]}, {-1, 0}];
epilog = Table[{a, t}, {p, {0.1, 0.5, 0.9}}];
DiscretePlot[Evaluate@Table[PDF[A,k], {p, {0.1, 0.5, 0.9}}], {k, 0, 15},
PlotRange -> All, PlotMarkers -> Automatic, Epilog -> epilog,
Background -> RGBColor[0.1, 0.45, 0.3, 0.9]]
a := {Arrowheads[0.02], Arrow[{{4, PDF[A, 0]}, {0, PDF[A, 0]}}]};
t := Text[Style["p=" <> ToString[p], Medium], {4, PDF[A, 0]}, {-1, 0}];
epilog = Table[{a, t}, {p, {0.1, 0.5, 0.9}}];
DiscretePlot[Evaluate@Table[PDF[A,k], {p, {0.1, 0.5, 0.9}}], {k, 0, 15},
PlotRange -> All, PlotMarkers -> Automatic, Epilog -> epilog,
Background -> RGBColor[0.1, 0.45, 0.3, 0.9]]
Expection and Variance:
If Y is a random variable with a geometric distribution,then
$$E(Y) = \frac{1-p}{p}\;\;\;{\rm{and}}\;\;\;V(Y) = \frac{{1 - p}}{{{p^2}}}.$$
You can use the following Mathematica command to obtain these results
Expectation[x, x \[Distributed] GeometricDistribution[p]]
or Mean[GeometricDistribution[p]]
Variance[GeometricDistribution[p]]
A Important Property:
Let Y denote a geometric random variable with probability of success p, $a$ is a nonnegative integer, then,
$$P(Y \ge a) = {(1 - p)^{a}}.$$
For nonnegative integers $a$ and $b$,
$$P(Y \ge a + b|Y \ge a) = {(1 - p)^b} = P(Y \ge b).$$
This property is called the memoryless property of the geometric distribution.
https://blog.sciencenet.cn/blog-531885-597408.html
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