2019-01-16[Stay Sharp]Bernoulli

作者: 三千雨点 | 来源:发表于2019-01-16 23:04 被阅读4次

Bernoulli distribution

Bernoulli distribution is the discrete probability distribution of a random variable which takes the value 1 with probability $u$ and the value 0 with probability $v = 1-u$ . i.e.
$p ( x = 1 | \mu ) = \mu$ where $0 \leqslant \mu \leqslant 1$ , and from that we get $v = p(x=0|\mu) = 1-\mu$ . so the probability distribution over $x$ can be writter in the following form:

$\operatorname { Bern } ( x | \mu ) = \mu ^ { x } ( 1 - \mu ) ^ { 1 - x }$
this is the Bernoulli distribution.

mean of Bernoulli distribution is $\mathbb { E } [ x ] = \mu$

Verification:

$\mathbb {E}[x] = p(x=1|\mu) * 1 + p(x = 0|\mu) * 0 = \mu * 1 + v * 0 = \mu$

variance of Bernoulli distribution is $\operatorname { var } [ x ] = \mu ( 1 - \mu )$

Verification:

$\mathbb { E } \left[ x ^ { 2 } \right] = p(x=1|\mu) * 1^{2} + p(x = 0|\mu) * 0^{2} = \mu * 1 + v * 0 = \mu$
$\operatorname {var}[x] = \mathbb { E } \left[ x ^ { 2 } \right] - \mathbb { E } [ x ] ^ { 2 } = \mu - \mu^{2} = \mu(1-\mu)$

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