A normal distribution in a variate X with mean mu and variance sigma^2 is a statistic distribution with probability density function P(x)=1/(sigmasqrt(2pi))e^(-(x-mu)^2/(2sigma^2)) (1) on the domain x in (-infty,infty). While statisticians and mathematicians uniformly use the ter…
The distribution function D(x), also called the cumulative distribution function (CDF) or cumulative frequency function, describes the probability that a variate X takes on a value less than or equal to a number x. The distribution function is sometimes also denoted F(x) (Evans e…
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This book is a concise presentation of the normal distribution on the real line and its counterparts on more abstract spaces, which we shall call the Gaussian distributions. The material is selected towards presenting characteristic properties, or characterizations, of the normal…
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The following theorem is proved. Let $X_1, X_2, \cdots, X_n$ be $n$ independently (but not necessarily identically) distributed random variables, and assume that the $n$th moment of each $X_i(i = 1, 2, \cdots, n)$ exists. The necessary and sufficient conditions for the existence…
In 2016, Karney proposed an exact sampling algorithm for the standard normal distribution. In this paper, we study the computational complexity of this algorithm under the random deviate model. Specifically, Karney’s algorithm requires the access to an infinite sequence of indepe…
Deconvolution problems arise in a variety of situations in statistics. An interesting problem is to estimate the density $f$ of a random variable $X$ based on $n$ i.i.d. observations from $Y = X + \varepsilon$, where $\varepsilon$ is a measurement error with a known distribution.…
A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range [0,x], Phi(x)=Q(x)=1/(sqrt(2pi))int_0^xe^(-t^2/2)dt. (1) It is related to the probability integral alpha(x)=1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (2)…
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