Questions tagged [probability-calculus]

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3
votes
1answer
274 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 ...
2
votes
1answer
43 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 ...
2
votes
1answer
40 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_{(...
5
votes
1answer
147 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 ...
4
votes
2answers
2k 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\...
0
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0answers
386 views

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 ...
14
votes
7answers
3k 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 ...
1
vote
0answers
77 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 ...
4
votes
1answer
256 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 ...
0
votes
1answer
165 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,...
3
votes
1answer
382 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; ...
1
vote
1answer
62 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)? $$\...
-1
votes
2answers
960 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 \...
4
votes
2answers
374 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 ...
0
votes
1answer
4k 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 ...
-1
votes
1answer
62 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 ...
1
vote
3answers
150 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 ...
1
vote
1answer
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-...