In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function.

This paper deals mainly with various ‘distribution functions’ and ‘cumulative distribution functions’ pertaining to the modulus and to the angle of the ‘normal’ complex variate, for the case where the mean value of this variate is zero.

Characteristic functions are essentially Fourier transformations of distribution functions, which provide a general and powerful tool to analyze probability distributions. Recall that in order to check convergence in distribution for a sequence of random quantities Xn, we need to show convergence of Ef(Xn) for all bounded continuous function f.

Mar 25, 2025 · 3. Cumulative Distribution Function (CDF): it indicates the cumulated probability up to a certain value, for example, the probability of a newborn baby weighing up to 2.5kg. It is denoted by F(x)=P[X≤x]. These three functions are visually represented through the …

Aug 26, 2015 · Let $X$ be a random variable with characteristic function $\phi(\ )$ satisfying $|\phi(t)|=1$ for all $|t|\leq 1/T$ with some $T>0$. Show that $X$ is degenerate, i.e., there is $c$ such that $P(X=c)=1$.

Dec 8, 2017 · There is a standard result in probability theory that whenever $X$ and $Y$ are independent, then so are $f(X)$ and $g(Y)$ for (measurable) functions $f$ and $g$. This yields independence directly. For the distribution you need to apply the …

distribution of X is usually represented by, X ∼ N(µ,σ2), or also, X ∼ N(x−µ,σ2). The Normal or Gaussian pdf (1.1) is a bell-shaped curve that is symmetric about the mean µ and that attains its maximum value of √1 2πσ ’ 0.399 σ at x = µ as represented in Figure 1.1 for µ = 2 and σ 2= 1.5 .

Jul 1, 2017 · This paper deals mainly with various ‘distribution functions’ and ‘cumulative distribution functions’ pertaining to the modulus and to the angle of the ‘normal’ complex variate, for the case where...

Apr 18, 2019 · Given that $X$ is uniformly distributed over $K$ (that is, the probability density function of $X$ is $f(x) = \frac{1}{2^{64}}$), what is the distribution of $Y = X \bmod k$? My gut says it's uniformly distributed over $M = [0, k)$ (i.e., the pdf of $Y$ is $f(y)=\frac{1}{k}$ ) but I can't quite sort the math to get me there.

Dec 8, 2013 · The characteristic function of a probability measure m on B(R) is the function jm: R!C given by jm(t) = Z eitx m(dx) When we speak of the characteristic function jX of a random vari-able X, we have the characteristic function jm X of its distribution mX in mind. Note, moreover, that jX(t) = E[eitX]. While difficult to visualize, characteristic ...