A Geometric experiment possesses the following properties: 1. The experiment consists of aseries 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 totrial. 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 failuresbefore asuccess 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 , y] Relative Mathematica Functions GeometricDistribution represents a geometric distribution with success probability p. Examples: A = GeometricDistribution ; a := {Arrowheads , Arrow }, {0, PDF }}]}; t := Text , Medium], {4, PDF }, {-1, 0}]; epilog = Table ; DiscretePlot , {p, {0.1, 0.5, 0.9}}], {k, 0, 15}, PlotRange - All, PlotMarkers - Automatic, Epilog - epilog, Background - RGBColor ] 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 GeometricDistribution ] or Mean ] Variance ] A Important Property: Let Y denote a geometric random variable with probability of success p, $a$ is anonnegative 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.
What is it? Geometric measure theory is concerned with investigating the structure of surfaces from a measure-theoretic viewpoint. Since the notion of a surface (in an appropriately general sense) appears in many different settings in mathematics, it is unsurprising that GMT has applications in many areas of modern mathematics including: PDEs, Harmonic analysis and variational problems. GMT is traditionally considered to be a hard subject. This is primarily because, although many of the ideas involved are simple, in order to work at the level of generality that we do, a lot of technicalities need to be considered and understood in order to prove useful results. To gain a brief overview of the subject, you may like to look at an article that the paper online from Springer.
标签: Geometric
