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^...
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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
$$β(a)...
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Prove that E[x^n] >= (EX)^n for n = 2k [duplicate]
Prove that $E[x^n] \geq (Ex)^n$ for $n = 2k$
I only have the formula of E(x) but I don't know how to prove it.
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Normal distribution - finding X value given probability P(-x < X < x) = 0.95
I am learning statistics by myself online and I just encountered a problem that I am not able to solve.
$X \sim N(10, 4)$. I need to find $P(-x<X-10<x) = 0.95$. It resulted to the following $P(Z&...
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Reason for absolute value of Jacobian determinant in change-of-variable formula?
When we have a random variable $x$ with a probability density $p(x)$, and a function $y = f(x)$ that is differentiable and can be solved for $x = g(y)$, the change of variable formula leads us to a ...
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Derivative of expectation where the variable appears in the integration limit and in the integrand?
I want to calculate the derivative of
$$\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 ...
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$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_{(...
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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; ...
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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)?
$$\...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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