All Questions
6 questions
3
votes
1
answer
2k
views
From Lebesgue Integral to Stieltjes Integral, and integration by parts
Let $X$ be a real random variable with c.d.f function $F$.
Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite).
What additional ...
- 591
3
votes
0
answers
228
views
Sub-multiplicative function in expectation or pointwise? [closed]
Consider the function that satisfies
$$ \mathbb{E}[f(X)f(Y)]\leq \mathbb{E}[f(XY)],$$
where $X\in\mathbb{R}$ and $Y\in\mathbb{R}$ are Gaussian random variables with mean $0$ and variance $1$, and ...
2
votes
2
answers
407
views
How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?
Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim ...
- 139
1
vote
1
answer
853
views
Quadrature methods for high-dimensional Gaussian integration
Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous ...
- 5,405
0
votes
0
answers
149
views
Reference book for a probability course
In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
- 1,561
-6
votes
2
answers
2k
views
Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
- 43.2k