Questions tagged [probability-calculus]

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20 questions
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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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Change in function value between correlated variables

Suppose i have dependent random variables $X_1,X_2,\ldots,X_n$ and a function $z = f(X_1,X_2,\ldots,X_n)$ and suppose i want to perturb one of the features $X_1$ by $\Delta X_1$ (i'm happy to use a ...
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Do the parameters that arise in de Finettis representation theorem follow the rules of probability?

I recently stumbled upon de Finettis (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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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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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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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 ...