Tagged Questions
2
votes
0answers
29 views
Integral of the Normal Characteristic Function
The characteristic function of the $N$-variate Normal distribution is
$$\forall \mathbf{t} \in \mathbb{R}^N \quad
\psi(\mathbf{t}) \equiv \mathbb{E}\left(
e^{i\mathbf{t}X}\right) =
\exp \left( i{ ...
1
vote
2answers
29 views
Bound for erf function
For small $\epsilon \geq 0$ Is
$erf(\epsilon) \leq \epsilon$
Can somebody give me the hint
1
vote
1answer
57 views
Normal Distribution Identity
I have the following problem. I am reading the paper which uses this identity for a proof, but I can't see why or how to prove its true. Can you help me?
\begin{align}
\int_{x_{0}}^{\infty} e^{tx} ...
2
votes
1answer
73 views
Techniques for evaluating probability integral
Consider the integral of a normal distribution:
$$\int_a^b f(x)\,\mathrm d x=c
$$
and a second integral for the expected value:
$$
\int_a^b x\cdot f(x)\,\mathrm dx
$$
Since you know the first ...
0
votes
1answer
70 views
How to show that the inverse Gaussian density integrates to 1?
How to prove
$\int_{0}^{\infty}\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp\left\{\frac{-\lambda(x-\mu)^2}{2\mu^2 x}\right\}dx=1$?
2
votes
0answers
101 views
Proof of a gaussian integral turning into a cosine
I have a numerical evidence of
$$\int_0^{1/2} \frac{1}{\sqrt{2\pi}\sigma_0x}\exp\left(-\frac{(\mu_0x-y)^2}{2\sigma_0^2x^2}\right)dx \approx 1+\cos(2\pi y),$$ where ...
3
votes
1answer
92 views
Is there a closed-form expression for the integral of this product of gaussian functions?
Considering:
$$f(x) = \frac{1}{\sigma_x\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x}{\sigma_x})^2}$$
$$g_i(x) = \frac{1}{\sigma_i\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{a_i+b_ix}{\sigma_i})^2}$$
Is there a ...
1
vote
1answer
589 views
Linear transformation of normal distribution
Not sure if "linear transformation" is the correct terminology, but...
Let $X$ be a random variable with a normal distribution $f(x)$ with mean $\mu_{X}$ and standard deviation $\sigma_{X}$:
$$f(x) = ...
0
votes
1answer
559 views
How to apply Central Limit Theorem to Uniform Distribution to generate Normal Distrubution?
Suppose I have a simple uniform continuous "unit" distribution X:
$$\begin{align*}
\forall y \in \mathbb{R} \implies \\
y < 0 : & P(X < y) = 0 \\
y \in [0,1] : & P(X < y) = y \\
...
1
vote
4answers
642 views
Quantile function with Normal distribution and Weibull distribution
A quantile function Q is defined in terms of its distribution function F as:
$Q(p)=inf\{ x\in R:p \le F(x)\},p\in(0,1)$
But i don't understand very well how it works exactly. Suppose we are managing ...
