All Questions
Tagged with integration probability
1,744
questions
7
votes
3
answers
878
views
This expected value has a minimum!
Problem. Let $X$ be a positive, real random variable whose probability density function is bounded by $1$. Prove that $E[X]\geq \frac 12$.
Hi everyone. This problem is essentially saying that the ...
- 984
1
vote
1
answer
53
views
Conditional expectation - alternative expression
Consider the following set-up.
$F:[0,\omega]\rightarrow[0,1]$ where $X$ is a real-valued random variable.
The conditional expectation of $X$ given $X<x$ is:
$E(X|X<x)=\frac{1}{F(x)} \int_0^s tf(...
- 1,467
5
votes
0
answers
62
views
Integrating the Beta Function
As a learning exercise, I am trying to find the mean and variance of the Beta Probability Distribution (https://en.wikipedia.org/wiki/Beta_distribution) from first principles (i.e. Method Of Moments):
...
- 399
-1
votes
0
answers
11
views
Expected value of the gradient of logistic loss function to a normal distribution
Given a vector $\beta\in\mathbb{R}^d$, and a random vector $x\sim\mathbf{N}(0_d,I_{d\times d})$, that is $\{x_j\}_{j=1}^d$ are i.i.d generated from gaussian $\mathbf{N}(0,1)$, can we compute or ...
- 1
2
votes
1
answer
61
views
An application of Chebyshev association inequality?
Let $X$ be a r.v and let $f \geq 0 $ be a nonincreasing function, $g$ be a nondecreasing
real-valued function. Suppose $h\geq 0$ is a function such that $h(X)$ has finite expectation with $E[h(X)f(X)]...
0
votes
0
answers
21
views
Help me calculate the assumed interest rate $i'$ under $\frac{\bar{A}_x}{\mu_x}=\bar{a'}_x \neq \bar{a}_x$
Setup
force of mortality $\mu_x = 0.005\cdot (1.01)^x$
original assumed interest rate $i = 0.05$
assumption:
$$
\frac{\bar{A}_x}{\mu_x}=\bar{a'}_x (\neq \bar{a}_x)
$$
At this point, we want to ...
- 590
0
votes
0
answers
13
views
Deriving the Variance of a Posterior Conditional Probability Density Function
Deriving the variance of a conditional probability density function
I am trying to derive the variance of a conditional probability density function $ p_{r_s|x,y_s} $ given by:
$$ p_{r_s|x,y_s} = \...
- 311
1
vote
0
answers
21
views
Solving the marginal likelihood integral and approximating closed-form solutions for that
Given the likelihood function:
\[ p_{y_s|r_s,x}(y_s|r_s,x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(y_s - \alpha \frac{\rho_s}{r_s^4})^2}{2}} \]
and the prior distribution for \( r_s \):
\[ p_{r_s}(r_s) = \...
- 311
0
votes
1
answer
44
views
Cumulative distribution function of chi-squared distribution
I am trying to understand the proof here. At some point goes from
$$F_Y(y) = \frac{F_Y(y)}{\mathrm{d}y} \mathrm{d}y = f_Y(y) \mathrm{d}y \tag{1}$$
to
$$f_Y(y) \mathrm{d}y = \int_{V} \prod_{i=1}^{k} \...
- 865
4
votes
2
answers
111
views
Does $E[E[X|X\le Y]]=E[X]$?
This is a question I came up with while doing some related calculations.
Let $X$ be a random variable defined on some probability space $(\Omega, \mathcal A, P)$. To make everything as nicely behaved ...
- 3,246
0
votes
2
answers
68
views
Prove $E(X) = a$ for a pdf symmetric about $a$ [closed]
Show that if $X$ is a continuous random variable with p.d.f. $f(x)$ such that $f (a + x) = f (a − x)$, then $E(X) = a.$
I did a change of variable and set $x=a+y$, but I'm having difficulty getting to ...
- 39
0
votes
1
answer
52
views
Working on details on the Secretary Problem [closed]
I've been trying to follow this proof of the optimal way to solve the secretary problem (ref. https://en.wikipedia.org/wiki/Secretary_problem). Everything is clear to me except where they are ...
- 142
1
vote
0
answers
21
views
How can we fill the integration bounds and define a function $s$ from the given data so that this double integral is the mean of this indefinite int?
Suppose $f:\mathbb R \to \mathbb R$ has $\int_{-\infty}^\infty f(t)\ dt = 1 \in \mathbb R$.
Define $g : \mathbb R \to \mathbb R$ by $g(\alpha) = \int_{-\infty}^\alpha f(t)\ dt$ and define $h : \mathbb ...
- 972
0
votes
0
answers
34
views
Smoothness of expectation of a piecewise function
Suppose $f(x)$ and $g(x)$ are piecewise smooth functions. For simplicity, we can assume that $f(x)$ has $m$ pieces, and $g(x):=\max_{i=1,2,\ldots, I}\left\{k_i~ x+b_i\right\}$.
I have two questions:
...
- 97
1
vote
0
answers
29
views
Gaussian line integral over a polygon
Problem definition
Let $y=z+v$, where $z$ is uniformly distributed over the contour of a polygon with vertices $V_1,\dots,V_n \in \mathbb{R}^2$ and $v\sim\mathcal{N}(0,R)$. Let $\ell$ be the set of ...
- 729