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On the other hand, a mixture density created as a mixture of two normal distributions with different means will have two peaks provided that the two means are far enough apart, showing that this distribution is radically different from a normal distribution. Mixture distributions arise in many contexts in the literature and arise naturally ...
The normal distribution plotted with different means and variances. Financial returns often behave differently in normal situations and during crisis times. A mixture model [3] for return data seems reasonable. Sometimes the model used is a jump-diffusion model, or as a mixture of two normal distributions.
A study of a mixture density of two normal distributions data found that separation into the two normal distributions was difficult unless the means were separated by 4–6 standard deviations. [59] In astronomy the Kernel Mean Matching algorithm is used to decide if a data set belongs to a single normal distribution or to a mixture of two ...
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution. The probability density function of a normal variance-mean mixture with mixing probability density g {\displaystyle g} is
The triangular distribution on [a, b], a special case of which is the distribution of the sum of two independent uniformly distributed random variables (the convolution of two uniform distributions). The trapezoidal distribution; The truncated normal distribution on [a, b]. The U-quadratic distribution on [a, b].
In probability theory and statistics, a mixture is a probabilistic combination of two or more probability distributions. [1] The concept arises mostly in two contexts: A mixture defining a new probability distribution from some existing ones, as in a mixture distribution or a compound distribution. Here a major problem often is to derive the ...
This follows immediately from the Laplace distribution resulting as a normal scale mixture; see above. The roles of conditional and mixing distributions may also be exchanged here; consequently, compounding a Rayleigh distribution with its scale parameter distributed according to a half-normal distribution also yields an exponential distribution.