# Questions tagged [integral]

For on-topic question related to uses of the mathematical concept of an integral, i.e. $\int_a^b f(x)\; dx$. Purely mathematical questions about integrals are better asked at math SE: https://math.stackexchange.com/

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### How to solve for $k$ in integral? (hypothesis testing)

In a hypothesis testing problem I am stuck at this step, trying to solve for $k$ in the relation: $$F_{NegBin(n,0.5)}(k)=0.05$$ The problem is no matter which cumulative distribution function I ...
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### Parameter in expectation subscript

There are a few questions/answers out there about subscripts involving a random variable ($E_X[...]$) or a density ($E_{f(X)}[...]$) in expectations. I like this one. But today I ran into a ...
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### What does “the denominator does not contain any theta dependence” mean in Bayes' Rule? [duplicate]

Every lecture and book says that the denominator in Bayes' Rule does not depend on the parameter $\theta$. However, the denominator also includes $\theta$ in the formula of Bayes' Rule. I just cannot ...
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### Distribution of squared Brownian Motion Conditional on its integral

I am interested in the distribution of $W_1^2$ conditional on $\int_0^1 W^2$. Simulations suggest the conditional mean is $\int_0^1 W^2-1/2$, and the variance is approximately one half that, so it ...
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### Evaluate an integral using importance sampling

Estimate $\int^{1}_{0}e^{x} dx$ using importance sampling. Should I use beta distribution as proposal distribution and uniform distribution as target ?
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### Symmetry between integrals including absolute value

So I came across below symmetry in my probability course that I can't understand. I understand how the lower bound changes when removing the absolute value operator, but how does the 2 disappear?
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### Closed form solution for integral involving inverse Mills ratio?

Is there a closed form solution for the integral $$\int_0^a \sqrt{c} \phi\left(\frac{\mu}{\sigma}\sqrt{c}\right) m e^{-mc} dc$$ where $\phi(x)$ is the standard normal pdf? If it exists what is it? I'...
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### Find conditional pdf given joint

Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$. What is the marginal cdf of $X$? of $Y$? Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
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### How to implement single Imputation from conditional distribution?

In [*] page 264, a method of drawing a missing value from a conditional distribution P(X_mis|X_obs;Theta) which is defined as: I did not find any code ...
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### Understanding Gauss-Hermite Weights

I routinely use Gauss-Hermite as a tool for approximating complex integrals. While I am proficient in its applications, I am not proficient in its development. I am working to understand the weights ...
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### Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

For a uniformly distributed variable between 0 and 1 generated using rand(1,10000) this returns 10,000 random numbers between 0 and 1. If you take the mean, it ...
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### How to obtain value range of empirical PDF, given the mode and area?

Given an empirical PDF of a continuous random variable $X$, then integrating over its entire defined domain will yield an area of size 1. To find the probability of $X \ge x_1 \land X \le x_2$ (as in ...
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### Choice probabilities and integration

I have a question from the book "Discrete Choice Methods with Simulation" (Train, 2009). In chapter 1, , part 1.2.3, partial simulation, partial closed form, it has the following equation which ...
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### Why do we integrate to obtain prior distributions?

In the test for difference in means with unknown variance, it's stated that in order to obtain prior distributions, we have to take the double integral with respect to mean and to the variance? ...
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### Integrated time series and continuous time integral

I am new to time series, and have a theoretical (philosphical, if you wish) question on the connection of an integrated time series with the usual concept of integration in calculus. If $x_t$ is an ...
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### Integration of (Phi(x)-Phi(y))^2d(F(x, y)

How to integrate the following? Thanks
for the last couple of days I have been struggling with a problem and I was hoping to get some help here. I have a function that looks as follows:  f(x,y,z)=\begin{cases} 0 & \text{if} \ (x*...