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<B> [[Expected Value|Expected Value]]</B>: The expected value of a [[random variable]] <math>X</math> is denoted by <math>E(X)</math> and may be interpreted as the long-term average value of <math>X</math> In the case of a [[discrete random variable]], taking values <math>x_l, x_2, .. . </math> | <B> [[Expected Value|Expected Value]]</B>: The expected value of a [[random variable]] <math>X</math> is denoted by <math>E(X)</math> and may be interpreted as the long-term average value of <math>X</math> In the case of a [[discrete random variable]], taking values <math>x_l, x_2, .. . </math> | ||
:<math>E(X) = \sum_jx_jP(X=x_j).</math> | :<math>E(X) = \sum_jx_jP(X=x_j).</math> | ||
For a [[continuous random variable]] | For a [[continuous random variable]] with [[probability density function]] 12 | ||
:<math>E(X) = \int_{-\infty}^{\infty}xf(x)dx.</math> | :<math>E(X) = \int_{-\infty}^{\infty}xf(x)dx.</math> | ||
The expected value of <math>X</math> is often referred to as the <B>expectation</B> of <math>X</math> or as the [[mean]] of <math>X</math>. The word 'expectation' has been used in this context since the use of ‘expectatio’ by ‘Huygens in his 1657 treatise on the results of playing games of chance: De Rariocinifs in Ludo Aleae. If <math>g</math> is any function, the [[expected value]] of <math>g(X), E[g(X)]</math>, is defined by | The expected value of <math>X</math> is often referred to as the <B>expectation</B> of <math>X</math> or as the [[mean]] of <math>X</math>. The word 'expectation' has been used in this context since the use of ‘expectatio’ by ‘Huygens in his 1657 treatise on the results of playing games of chance: De Rariocinifs in Ludo Aleae. If <math>g</math> is any function, the [[expected value]] of <math>g(X), E[g(X)]</math>, is defined by | ||
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<B> [[Exponential distribution|Exponential distribution]] ([[negative exponential distribution|negative exponential distribution]])</B>: A [[random variable]] <math>X</math> with an exponential distribution has [[probability density function]] | <B> [[Exponential distribution|Exponential distribution]] ([[negative exponential distribution|negative exponential distribution]])</B>: A [[random variable]] <math>X</math> with an exponential distribution has [[probability density function]] f given by | ||
:<math>f(x) = \lambda e^{-\lambda x}, {x \geq 0},</math> | :<math>f(x) = \lambda e^{-\lambda x}, {x \geq 0},</math> | ||
where <math>\lambda</math> is a positive constant. The distribution function F is given by | where <math>\lambda</math> is a positive constant. The distribution function F is given by | ||
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<B> [[Exponential family|Exponential family]]</B>: If a [[random variable]] X has a [[probability distribution]] (*discrete case) or a [[probability density function]] | <B> [[Exponential family|Exponential family]]</B>: If a [[random variable]] X has a [[probability distribution]] (*discrete case) or a [[probability density function]] [*continuous case) that can be written in the form | ||
:<math>exp\{a(x)b(\theta) + c(\theta) + d(x)\}</math> | :<math>exp\{a(x)b(\theta) + c(\theta) + d(x)\}</math> | ||
where <math>\theta</math> is a parameter and a, b, c, and d are known functions, then it is a member of the exponential family. The [[Poisson distribution]], the [[binomial distribution]], and the [[Normal Distribution|normal distribution]] | where <math>\theta</math> is a parameter and a, b, c, and d are known functions, then it is a member of the exponential family. The [[Poisson distribution]], the [[binomial distribution]], and the [[Normal Distribution|normal distribution]] are members of this family. If <math>a(x) = x</math>, then the distribution is in canonical form and <math>b(\theta)</math> is a natural parameter of the distribution. | ||
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