All Questions
25
questions
1
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39
views
Is every probability mass function $f_X$ of a random variable $X$ the Radon-Nikodym derivative of $X_*P$ with respect to the counting measure?
Let $X$ be a discrete random variable (meaning $\text{im}(X)$ is countable) from a probability triple $(A,\mathcal{A},P)$ to a measurable space $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the ...
- 5,166
0
votes
0
answers
11
views
Integral of a function with respect to a measure on a mixed space $[\![ N ]\!]\times \mathsf{X}$
Task
$\newcommand{\intbrackets}[1]{[\![ #1 ]\!]}$
$\newcommand{\Pcal}{\mathcal{P}}$
I would like to better understand integration of a function $f$ with respect to a probability measure $\mu$ on a ...
- 5,247
1
vote
2
answers
92
views
How to generalise inner product to measures without densities
Let $(E, \mathcal{E}, \lambda)$ be a metric finite measure space, and let $\mu, \nu$ be finite measures with densities $f,g$ with respect to $\lambda$.
Then, I am interested in considering the ...
- 2,466
2
votes
1
answer
125
views
Reconciling uniform integrability definitions and their relation to tightness
Take a family $F$ of measurable functions on $\mathbb{R}$, then $F$ is uniformly integrable if:
Definition 1 (Royden): For each $\epsilon >0$, there is a $\delta>0$ such that for each $f\in F$, ...
- 529
1
vote
0
answers
19
views
On the sum of i.i.d. $(g_n)_{n=1}^\infty$
Let $(\Omega, \mathcal{A}, P)$ be a probability space.
Let $(g_n)_{n=1}^\infty$ be i.i.d. And let $g_n:\Omega \to \mathbb{R}$ be a random variable that follows the standard normal distribution.
It is ...
- 627
2
votes
2
answers
73
views
Where's my mistake in my attempt at showing that the squared sum of normally distributed variables is a $\chi^2$ distribution?
$\newcommand{\N}{\mathcal{N}}\newcommand{\d}{\,\mathrm{d}}$I am trying to understand how the sum of squares of standard normal variables is a $\chi^2$ distribution. Note that I am not a student of ...
- 42.7k
0
votes
1
answer
42
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Integration: Defining the integrand as a part of the measure
Let $P$ be a probability measure supported on $\mathbb{R}$. I am integrating:
$$\int_{x \in \mathbb{R}} c(x) \mathrm{d} P(x)$$
for some $c:\mathbb{R} \mapsto \mathbb{R}$. I was wondering the extremely ...
1
vote
0
answers
49
views
Integrating wrt a measure that is a collection of probability measures
Let $\mu$ be a probability measure defined on $\mathcal{A} \times \mathbb{R}$ where $\mathcal{A}$ is some measurable set and $\mathbb{R}$ is the set of real numbers. We have that $\mu(a, \cdot)$ is a ...
1
vote
2
answers
57
views
Given independent random variables $X$ and $Y$ with $f_X=\mathcal{X}_{[0,1]}$ and $f_Y=\frac{1}{2} \mathcal{X}_{[0,2]}$, find $P(Y> X).$
I am currently stuck on a computational problem for my measure theory course and wanted some feedback. The problem is as follows:
Given independent random variables $X$ and $Y$ with $f_X=\mathcal{X}_{...
- 2,295
1
vote
0
answers
89
views
Solving integral equation for a measure
For a given $a \in \mathbb R_{+}^*$, can we find a (positive) measure $\nu$ with support included in $\mathbb R$ such that, $\forall t \in \mathbb R_+^*$, $$at^2 = \int\limits_{\mathbb R} \ln\left(\...
- 286
5
votes
1
answer
71
views
If $\mathbb E_{\mathbb P} \vert f(X,Y)\vert <\infty$, is also $\mathbb E_{\mathbb P_1\times \mathbb P_2} \vert f(X,Y)\vert <\infty$?
Let $(E_1,\mathcal E_1)$ and $(E_2,\mathcal E_2)$ be two measurable spaces. Let $(E=E_1\times E_2,\mathcal E=\mathcal E_1\times \mathcal E_2)$. Let $\mathbb P$ a valid probability distribution on $(E,\...
- 528
1
vote
2
answers
56
views
Tight upper bound on the growth of the integral for $n$-th moment of normal distributions with increasing mean $\theta \to \infty, n\to \infty?$
Consider the growth rate integral:
$$I(\theta,n):= \int_{0}^{\infty}s^{n}e^{-(s-\theta)^2}ds, \theta > 0.$$
Thanks to the answer given by Kavi Ram Murthy, $C_n := \frac{I(\theta,n)}{\theta^n}$ ...
- 2,364
22
votes
3
answers
998
views
Derivative and calculus over sets such as the rational numbers
I am interested in the derivative of a function defined on a subset $S$ of $[0, 1]$. The subset in question is dense in $[0, 1]$ but has Lebesgue measure zero. My actual question can be found at the ...
- 3,645
1
vote
0
answers
40
views
Put-call parity for equity and debt share
Considering Merton's structural approach for credit risk modeling, we arrive to prove that the pricing formules are $S_t=V_t\phi(d_{T,1})-Fe^{-r(T-t)}\phi(d_{T,2})$ for equity share and $F_t=FP_0(t,T)-...
- 565
0
votes
1
answer
840
views
How to integrate wrt (dx)^2
Let us suppose I am taking some measurements whose outcomes are distributed according to a gaussian probability density distribution centered at $0$. For each measurement result falling in the range $\...
- 182