0
votes
1answer
38 views

Integral of cumulative distribution function

Yesterday I stumbled upon this question: For a distribution function $F(x)$ and constant $a$, integral of $F(x + a) - F(x)$ is $a$. It is basically proved that if $F(x)$ is a cumulative distribution ...
6
votes
6answers
206 views

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 ...
1
vote
3answers
42 views

Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
0
votes
2answers
34 views

Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
0
votes
1answer
24 views

Finding the boundaries of integration when calculating P(X + Y > a) or P(X + Y < b) (Jointly Distributed Continuous Random Variables)

I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$. For example, when I have $f(x,y) = \frac {x} {5}\ +\frac ...
0
votes
0answers
36 views

Urgent Find the CDF of U [duplicate]

I am having problems with Part 2. I know the upper limit of Y is x+u in the formula. But what about the limits of X ? Please help me !!
0
votes
0answers
22 views

PDE and probability integral

I have the following PDE $$\partial_tp(x,y,t)=-f(x,y,t)p(x,y,t)-g(x,y,t)\partial_xp(x,y,t)-h(x,y,t)\partial_yp(x,y,t)+D\partial_{xx}p(x,y,t)$$ where $f=2(1+a)x-3ax^2+1$, $g=x(x-1)(1-ax)-y+I+2x_j$, ...
1
vote
1answer
40 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
1
vote
0answers
113 views

What is the solution of the integral (product of two standard normal CDFs)?

I need to compute this kind of integral: where $b>0,d>0,a,c$ and $e$ are known constants and $\Phi$ is the CDF of the standard Normal distribution.
3
votes
0answers
851 views

Difficult integral for a marginal distribution

I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral: $$p(y)=\int_0^{\frac{1}{\sqrt{2 \pi }}} ...
5
votes
6answers
376 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
4
votes
2answers
101 views

Does anyone know how to calculate the following integral?

Consider the function (coming from a joint probability density): $$ f(x,y) = \frac{1}{y}e^{-y-\frac{x}{y}}. $$ I want to evaluate the definite integral (marginal): $$ F(x) = \int_0^\infty f(x,y)\,dy. ...
0
votes
1answer
29 views

Given 2 values for $P(X \gt A)$, and that x follows the Gaussian PDF, find expected value and standard deviation

Given that $P(X \gt 4) = 10^{-2}$ and that $P(X \gt 5) = 10^{-5}$ and that the random variable $X$ follows the Gaussian PDF, find the expected value and the standard deviation. To solve the above, ...
1
vote
1answer
51 views

Joint To Marginal Density : Can't figure it out.

Here goes the problem: Problem: Suppose $X$ and $Y$ have the joint density function: $f(X,Y) = c \sqrt{1 - x^2 - y^2}, \,\,\,\,\, x^2 + y^2 \leq 1$ Find $c$. ...
0
votes
0answers
28 views

Can this system of equations be simplified?

I have a system of two equations over $x$ and $y$: \begin{cases} \lambda \exp(2x\mu_x) \Phi(-y+\mu_y) = \Phi(-y-\mu_y) \\ \lambda \exp(2y\mu_y) \Phi(-x+\mu_x) = \Phi(-x-\mu_x) \end{cases} where the ...
1
vote
2answers
159 views

Integrating a special skew normal — the CDF of a convolution of a normal with a truncated normal

I am having a little trouble trying to compute an integral. In short, I wish to solve the following: $$F(x) = \int_{-\infty}^x \phi(au-b)\,\Phi(au+b)\,du $$ My intuition is that this might be ...
1
vote
0answers
114 views

Simplify the expectation of the maximum of two random variables

My aim is to simplify the maximum of two expressions each of which are a function of exponentially distributed random variables Given: positive constants $a,b,c,d$. Independent random variables $x,y ...
1
vote
2answers
143 views

Integrating exponential of exponential function

I would like to find the integral of $\int_0^\infty\exp(-u-\exp(-ku))\,du$ for $k>0$. This is related to the gumbel distribution(http://en.wikipedia.org/wiki/Gumbel_distribution), which shows ...
0
votes
1answer
47 views

Marginal distribution of $P$

Joint distribution of $P$ & $Q$ is $$f_{P,Q}(p,q)=\frac{1}{2\sqrt{(2\pi)}\sigma}\exp[-\frac{1}{2}{(\frac{\frac{p+q}{2}-\mu}{\sigma})^2}] \times\theta\exp[-\theta(\frac{p-q}{2})],\quad ...
3
votes
2answers
405 views

Characteristic Function of Inverse Gaussian Distribution

The pdf of Inverse Gaussian distribution, IG$(\mu,\lambda)$, is : $$p_X(x)=\sqrt\frac{\lambda}{2\pi x^3}\exp\left[\frac{-\lambda}{2\mu^2x}(x-\mu)^2\right];\quad x>0,\lambda,\mu>0$$ I have to ...
0
votes
1answer
945 views

Cumulative Distribution Function of Logistic Distribution.

pdf of logistic distribution is : $$p_X(x)=\frac{\pi}{\sigma\sqrt 3}\frac{\exp[\frac{-\pi(x-\mu)}{\sigma\sqrt3}]}{(1+\exp[\frac{-\pi(x-\mu)}{\sigma\sqrt3}])^2};\quad-\infty<x<\infty$$ I have ...
4
votes
1answer
970 views

Integral involving the CDF of normal distribution

I am doing some research and got stuck in solving the following integral (which I am not sure whether it has a closed form solution or not, I hope it has:)) Here is the integral: ...
0
votes
1answer
46 views

Please help finishing the calculation to find the Entropy of Pareto distribution.

Let $X$ follow Pareto distribution with parameters $\alpha, a, h$. That is, $X\sim Pa(\alpha,a,h)$, where $\alpha>0$ is the shape parameter, $-\infty < a < \infty$ is the location parameter, ...
1
vote
0answers
64 views

Please help finishing the calculation to prove that ” Pareto distribution & Power distribution has inverse relationship”.

Let X follows Pareto distribution with parameters α, a, h. that is X~Pa(α,a,h) Where, α>0 is the shape parameter, -∞< a <∞ is the location parameter, h>0 is the scale parameter. ...
3
votes
1answer
294 views

Integral of product of normal cdf and pdf

What do you think, is there a closed form solution of the following Integral $\textbf{ }$ $$\int_{-\infty}^{a-y}n(x)\, N(b-2y-x)\, dx,$$ where $N(x)=\int_{-\infty}^x n(z)\, dz\quad$ and $\quad ...
2
votes
1answer
230 views

Covariance of Student's t-distribution

Another integral (this time it looks like a lot of work but maybe it can be simplified). I have the Student's t-distribution $$\int_{-\infty}^\infty ...
1
vote
1answer
96 views

Solving $\int_0^\infty \left(1+\frac{y^{2}_{1}+y^{2}_{2}+\cdots+y^{2}_{n})}{\nu}\right)\mathrm dy_{1}\mathrm dy_{2}\cdots \mathrm dy_{n}$

Interesting integral here: $$\int_0^\infty \frac{\Gamma(D/2+\nu/2)}{\Gamma(\nu/2)}\frac{|\Lambda|^{1/2}}{(\pi \nu)^{D/2}}\left(1+\frac{(x-\mu)^{T}\Lambda(x-\mu)}{\nu}\right)^{-D/2-\nu/2}\mathrm dx$$ ...
2
votes
1answer
359 views

Computing an integral involving standard normal pdf and cdf - with peculiar limits.

I have had a look at some of the other questions on this topic but cannot quite work out the solution to this integral (or prove that there isn't a solution). Is there a way to work out: ...
1
vote
1answer
72 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
2
votes
2answers
205 views

Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: ...
0
votes
1answer
185 views

CDF integration question

I am solving a problem where $X$ is an exponential random variable and $\lambda=\frac{1}{10}$. I need to find the CDF of $X$ and have that $\int_0^\infty \frac{e^\frac{-x}{10}}{10}$ turns out to be ...
14
votes
2answers
292 views

How does one prove $\int_0^\infty \prod_{k=1}^\infty \operatorname{\rm sinc}\left( \frac{t}{2^{k+1}} \right) \mathrm{d} t = 2 \pi$

Looking into the distribution of a Fabius random variable: $$ X := \sum_{k=1}^\infty 2^{-k} u_k $$ where $u_k$ are i.i.d. uniform variables on a unit interval, I encountered the following ...
1
vote
2answers
145 views

Question on the standard normal distribution.

Let $X$ be a random variable having standard normal distribution. Let $\Phi$ denote its distribution function. Find $$ \int_0^\infty \operatorname{Prob} (\Phi(X) \geq u) \; du $$
5
votes
1answer
252 views

expectation of function of exponential

I need to compute the following integral $$ \int_{0}^{\infty}x\,\left\{\vphantom{\LARGE A}% 1- \left[\vphantom{\Large A}1- \exp(-a\,x^{\alpha}) \right]^M \right\}\,{\rm d}x \qquad \mbox{with}\quad ...
1
vote
2answers
349 views

Definite integral of cdf of the form $\Phi(\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}})$

Any solution for the following definite integaral? Here $\Phi(x)$ represents the cumulative distributive function of standard normal distribution ...