# Questions tagged [probability-calculus]

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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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### 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 ...
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 ...
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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 ...
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 ...
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,...
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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 ...
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 ...
62 views

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 ...