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The Bernoulli distribution is a special case of the binomial distribution with = [4] The kurtosis goes to infinity for high and low values of p , {\displaystyle p,} but for p = 1 / 2 {\displaystyle p=1/2} the two-point distributions including the Bernoulli distribution have a lower excess kurtosis , namely −2, than any other probability ...
When two random variables are statistically independent, the expectation of their product is the product of their expectations. This can be proved from the law of total expectation : E ( X Y ) = E ( E ( X Y ∣ Y ) ) {\displaystyle \operatorname {E} (XY)=\operatorname {E} (\operatorname {E} (XY\mid Y))}
There are cases in which uncorrelatedness does imply independence. One of these cases is the one in which both random variables are two-valued (so each can be linearly transformed to have a Bernoulli distribution). [3]
A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the binomial test of statistical significance. [1]
Then X 1 has the Bernoulli distribution with expected value μ = 0.5 and variance σ 2 = 0.25. The subsequent random variables X 2, X 3, ... will all be distributed binomially. As n grows larger, this distribution will gradually start to take shape more and more similar to the bell curve of the normal distribution.
The geometric distribution is the discrete probability distribution that describes when the first success in an infinite sequence of independent and identically distributed Bernoulli trials occurs. Its probability mass function depends on its parameterization and support .
This is not differentiable at t = 0, showing that the Cauchy distribution has no expectation. Also, the characteristic function of the sample mean X of n independent observations has characteristic function φ X (t) = (e −|t|/n) n = e −|t|, using the result from the previous section. This is the characteristic function of the standard ...
For example, a sequence of Bernoulli trials is interpreted as the Bernoulli process. One may generalize this to include continuous time Lévy processes, and many Lévy processes can be seen as limits of i.i.d. variables—for instance, the Wiener process is the limit of the Bernoulli process.
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