Tagged Questions

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How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
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Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt,$$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
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How to show this integral equals $\pi^2$?

I was trying to evaluate an integral related to the product of two cauchy distributions and in one of the steps got stuck in the integral $$\int_0^{\infty} \frac{\ln(x)}{\sqrt{x}(x-1)} dx.$$ I ...
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An Integral and its limit

Consider the following integral, $$K(\alpha)=\int_\mathbb{R}\log^2(g/f)(g/f)^\alpha f \, \mathrm{d}\mu$$ where $\alpha\geq 0$, $\,\mu$ is some measure and $f,g$ are some distinct continuous ...
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Does the symmetry of density functions in the integral equation imply another symmetry?

For two given distinct density functions $f_0$ and $f_1$, with $f_1(y)=f_0(-y)\,\forall y$, the following relation is known to hold for any $\alpha\in\mathbb{R}$ and $x$: \begin{align} x&= ...
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Calculus Question: Improper integral $\displaystyle\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$

I am curious about evaluation of the following integral $$\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$$ Is it possible to evaluate it? This not my homework but I will share my attempt. I tried ...
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tricky integrating ranges x1-x2

So, we know the sum of n i.i.d. exponential(lambda) is gamma(n,lambda). But I am looking at a problem with X1-X2. So I get the joint dist of z=x1-x2 and w=x2. Then I integrate out w on range 0 to ...
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Integral of a bivariate normal cdf

Let $$\Phi_2(x,y;\rho):=\int_{-\infty}^y\int_{-\infty}^x \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(s^2+t^2-2st\rho)} \, ds \, dt$$ be the joint cdf of bi-variate normal random ...
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A limit of an Integral

Consider the following limit $$K=\lim_{x\rightarrow \infty}\frac{1}{x(1-x)}\left(1-\int_{\mathbb{R}}g(y;x)^x f(y)^{1-x}\mathrm{d}y\right)$$ where $f$ and $g$ are any continuous probability density ...
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Need an example/counterexample of continuous and increasing function.

If $\mu$ is a finite measure on the measurable space $\big( X, \mathscr{F} \big)$, $f : X\to [ 0, +\infty)$ is measurable. Then $\textbf{does it exist a continuous function$g : [ 0, +\infty)\to [ ...
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What is the pdf of $X,Y$?

We know that the common pdf of $X,Y$ is constant function, on the triangle $(0,0),(0,1),(2,0)$ (and out of this range the value of the function is zero). What is $f_X(x)$ and $f_Y(y)$? My solution: ...
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Joint distribution: show the components of the joint distribution are independent.

Very odd question I think... Show that if $(X,Y)$ is a random vector in $\mathbb{R}^{2}$ with density $f_{(X,Y)}(x,y) = f(x)g(y)$ for a pair of non-negative functions $f$ and $g$, then $X$ has ...
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Resolve integral with importance sample Monte Carlo

I'm trying to compute the integral $$\int_{a}^{b}(\sin( 1 + x ) + \cos( 1 + x ))e^{-x}\ dx$$ using importance sample Monte Carlo method. The exercise ask to use Cauchy Distribution to resolve the ...
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Finding conditionally expected $y$ given a specific $x$ from a joint distribution function!

I want to determine expected $y$, given $x=2$ given joint pdf shown below $$\frac{1}{2y} * e^{-\frac{y^2 + \frac{x}{2}}{y}}$$ for $x,y \gt 0$ and $0$ otherwise I believe this means I want ...
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Integrating the product of Poisson and exponential pdf

So I'll spare the background as to why, but I'm trying to integrate the following: $$\int_0^{\infty} \frac{e^{-(\lambda+\mu)t}(\lambda t)^n}{n!} dt$$ If you parameterize a Poisson w/ $\lambda$ and ...
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Gram-Charlier expansion, option price, higher derivative and integration by parts

I am currently reading a finance paper of Backus et al. (2004), called 'Accounting for biases in Black-Scholes'. To explain an abnormality called 'volatility smirk' that can be found in option prices, ...