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Questions tagged [probability-calculus]

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Problem with the expectation of transformed random variable [duplicate]

I read a paper where the random variable $z$ is assumed to follow a log-normal distribution. \begin{equation} \mathbb{E}\left(z^{1-\chi}\right)=\int_{0}^{\infty}z^{1-\chi}\frac{1}{z\sqrt{2\pi s^{2}}}e^...
optimal control's user avatar
1 vote
0 answers
82 views

How to calculate the coefficient of absolute regularity?

I am reading the paper and try to understand the example solution in p. 14. In particular, if the Markov chain has stationary distribution $\pi$ and $a$-step transition distribution $P^a$, then $$β(a)...
Nick's user avatar
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1 vote
1 answer
385 views

Prove that E[x^n] >= (EX)^n for n = 2k [duplicate]

Prove that $E[x^n] \geq (Ex)^n$ for $n = 2k$ I only have the formula of E(x) but I don't know how to prove it.
Ann Height's user avatar
0 votes
2 answers
2k views

Normal distribution - finding X value given probability P(-x < X < x) = 0.95

I am learning statistics by myself online and I just encountered a problem that I am not able to solve. $X \sim N(10, 4)$. I need to find $P(-x<X-10<x) = 0.95$. It resulted to the following $P(Z&...
Andrew's user avatar
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10 votes
1 answer
2k views

Reason for absolute value of Jacobian determinant in change-of-variable formula?

When we have a random variable $x$ with a probability density $p(x)$, and a function $y = f(x)$ that is differentiable and can be solved for $x = g(y)$, the change of variable formula leads us to a ...
Durden's user avatar
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2 votes
1 answer
706 views

Derivative of expectation where the variable appears in the integration limit and in the integrand?

I want to calculate the derivative of $$\varphi(\mu) = \int_{-\infty}^{\mu} r(x-\mu) f(x)dx,$$ wrt to $\mu$, where $r$ is a function and $f$ is a density function. How can I account for the presence ...
Martell's user avatar
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2 votes
1 answer
54 views

$k$-th order statistics when the value of $j$-th one is known

Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$. Suppose also that we know one of the realization, say $X_{(j)}=x_{(...
Andeanlll's user avatar
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5 votes
1 answer
234 views

Do the parameters that arise in de Finetti`s representation theorem follow the rules of probability?

I recently stumbled upon de Finetti`s (pretty cool) representation theorem (What is so cool about de Finetti's representation theorem?). I wondered whether the RV $\Theta$ that arises in this ...
Sebastian's user avatar
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4 votes
2 answers
6k views

Using Chebyshev's inequality to obtain lower bounds

Let $X_1$ and $X_2$ be i.i.d. continuous random variables with pdf $f(x) = 6x(1-x), 0<x<1$ and $0$, otherwise. Using Chebyshev's inequality, find the lower bound of $P\left(|X_1 + X_2-1| \le\...
Shreya Bhandari's user avatar
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Taylor expansion in xgboost [duplicate]

I'm reading through the math of xgboost: https://xgboost.readthedocs.io/en/latest/model.html Under the ADDITIVE TRAINING section of the objective function, I saw that in the derivation of the ...
clueless_undergrad37's user avatar
14 votes
7 answers
7k views

Intuitively understand why the Poisson distribution is the limiting case of the binomial distribution

In "Data Analysis" by D. S. Sivia, there is a derivation of the Poisson distribution, from the binomial distribution. They argue that the Poisson distribution is the limiting case of the binomial ...
Ytsen de Boer's user avatar
1 vote
0 answers
86 views

Solve probability equation for one variable

Given: p = $a^l$ * ($\frac{c}{a^l + b^l}$ + $\frac{b}{a^l + c^l}$) Where a, b, c, p are known and are probabilities. Solve for l. (1 equation and 1 unknown) Does a closed form solution to this ...
Savino's user avatar
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4 votes
1 answer
525 views

support of an importance sampling with respect to the original distribution function

I have a question regarding the support of an importance sampling distribution with respect to the support of the original distribution function. I was reading that the support of the importance ...
john_w's user avatar
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1 vote
1 answer
350 views

Probability mass function from multiple datasets with different ranges

Assume 3 datasets (experiments) for the same population of a discrete random variable, dataset 1 has observed values {1, 2, 3, NA} values, dataset 2 {1, 2, 3, 4, 5, NA} and dataset 3 {1, 2, 3, 4, 5, 6,...
user90772's user avatar
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3 votes
1 answer
595 views

Integral formula using R

I have to implement this formula: $K(x) = \int_{0}^{0.5}q_{\theta}(x)d{\theta}$ where $q_{\theta}(x)$´s are the conditional quantiles in some $\theta$. using a range of $\theta = [0.45; 0.40; 0.35; ...
cassius's user avatar
  • 253
1 vote
1 answer
91 views

Interchanging limit and derivative for CDFs

Let $F_{\theta}(x)$ denote a cumulative distribution function indexed by the parameter vector $\theta$. Given this definition is the following equation correct (and if so under which conditions)? $$\...
user304347's user avatar
-1 votes
2 answers
2k views

E(x) for uniform distribution

I would like to take a uniform distribution for example. Let’s say a train will arrive at the station randomly within every 10 minute window, so the probability density function is $f(t)=0.1$,$\: t \...
whoisit's user avatar
  • 737
5 votes
2 answers
1k views

Expected value of a function including the cumulative normal distribution

Consider the function $Y = 1 - 2 \Phi((c_j - \mu)/\sigma) + 2 \Phi^2((c_j - \mu)/\sigma)$ where $\Phi$ is the cumulative distribution function for the standard normal distribution and $c_j$ is a ...
user2728808's user avatar
0 votes
1 answer
12k views

If the probability density function of X is f(x)=sin(x) then what is the variance of X?

I am studying a random variable X with probability density function f(x) = sin(x) the domain of X is (0,pi/2). I'm trying to determine the variance of X. I know var(X) = E(X^2) - E(X)^2 how do ...
linksys's user avatar
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-1 votes
1 answer
67 views

Truly statistical task [duplicate]

I have a specific task in programming but I am curious about how long it will take to complete. I have an array "A" with 1000 unique numbers inside. For each iteration I am copying 30 randomly ...
Oskar's user avatar
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1 vote
3 answers
189 views

How a statistical package like SAS analyses market risk without any calculus support

SAS is a very popular tool to analyze the market risks of a portfolio of stocks,bonds including non linear components like options. Assuming that option analysis uses advanced calculus and even stock ...
Victor's user avatar
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1 vote
1 answer
2k views

What does correlation mean in error propagation?

From the python uncertainties package: Correlations between expressions are correctly taken into account. Thus, x-x is exactly zero, for instance (most implementations found on the web yield a non-...
naught101's user avatar
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