Tagged Questions
0
votes
0answers
52 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 ...
0
votes
0answers
10 views
Multiplication of Multivariate T distributions
I need to integrate two (unnormalised) multivariate T-distributions $\int t_{\nu_1}(x|0,C_1)t_{\nu_2}(x|\mu,C_2) dx $. Note that they are of two different degrees of freedom which makes the question ...
2
votes
2answers
42 views
integrating using student t distribution
Evaluate the integral
$\int_0^\infty\frac{1}{1+x^2}dx$
using the Student t distribution.
I don't know where to start. I am assuming that I can't just do regular integration. I don't know how I am ...
0
votes
0answers
48 views
Finding marginal distribution from joint distribution when domain of one variable is given in terms of the other variable
Given distributions $f_X(x)$ and $f_{Y \mid X}(y \! \mid \! x)$:
My aim is to find $f_Y(y)$.
I find the joint density as
$$ f_{XY}(x,y) = f_{Y \mid X}(y \! \mid \! x) f_X(x) = 0.5(y-x) \quad ...
1
vote
2answers
44 views
Integral question related to i.i.d. random variables
I am stuck on this question, I would appreciate any
hint I can get to understand this!
We have $(\Omega, M, P)$ a probability space. $X_1$ and $X_2$
are i.i.d. and let $S_2 = X_1 + X_2$.
For given ...
5
votes
0answers
223 views
Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)
As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
2
votes
1answer
109 views
Prove or disprove that the given expression is “always” positive
I have previously asked a question and I tried to solve it by my own and it led to the question below:
Prove or disprove that
...
1
vote
1answer
56 views
$e^{F(x,y)}$ Type Multi-variable Exponential Integrals
I am sure all you integration buffs can do this faster than I can type it. Your help with a quick explanation and solution is appreciated.
$$F _{XY} = \int_0^\infty\int_0^\infty xye^{-\frac{x^2 + ...
1
vote
1answer
60 views
Integrating out $\sigma^2$
My question is with regards to integrating on $\sigma^2$ when working with normal distributions. How does one handle the $^2$ aspect of $\sigma^2$?
I am unsure whether I should treat $\sigma^2$ as an ...
3
votes
0answers
104 views
An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.
We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
0
votes
2answers
33 views
Probability, Random points in rectangle
There is a rectangle, the lower left is always fixed at co-ordinate $(0, 0)$. Let the width and height of the rectangle be $w$ and $h$. Let $P$ be a randomly chosen point from the rectangle with ...
2
votes
1answer
153 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:
...
3
votes
1answer
123 views
Determining boundaries of Probability Density Function integral for a requested probability
This isn't one specific homework question, but a concept I'm having trouble with in class. We were asked on a couple of questions recently on homework dealing with the probability density function of ...
3
votes
2answers
138 views
Finding PDF involving absolute value
I'm trying to solve the following question:
Given an exponential R.V. X with rate parameter $\lambda > 0$, find the PDF of $V=|X-\lambda|$.
In order to find the PDF, I would like to use the CDF ...
1
vote
0answers
103 views
Numerical integration of binomial pdf with respect to a conditional probability
How do you numerically integrate the following loss function with respect to $\Phi(f_t)$, given $N_t=50$, $d_t=0$, $\rho=0.2$ and $\pi=0.01$?
$P(D_t=d_t)=\bigl(\begin{smallmatrix}
N_T \\
d_t
...
0
votes
1answer
46 views
Change of Variables and independent random variables.
Suppose that we have two IID random variables, $X_1, X_2$, carried by a triple $(\Omega,\mathcal{F},P)$.
While solving an exercise I ended to a point that I had to see that,
$$
\iint\limits_D x_1 ...
0
votes
1answer
43 views
simplifying an expression involving an integral
Simplify the following expression
$$
\iint_{-\infty}^{c+x}xf(x)f(y) \,dy\,dx+\iint_{c+x}^{\infty}yf(x)f(y) \,dy\,dx
$$
where $x$ and $y$ are iid random variables; $c$ is a constant; and $f$ is the ...
0
votes
1answer
77 views
Interpreting an integral/ probability
Think of two iid random variables $x$ and $y$ with density $f$ and CDF $F$ and a constant $c$. What could the qualitative meaning of the following expression be?
...
1
vote
3answers
157 views
What is wrong with this approach to find Expected value of distance between $X, Y \in$ Uniform $(0, 1)$?
Let $X$ and $Y$ be i.i.d. random variables with Uniform $(0, 1)$ continuous distribution.
The problem is to find the expected value of the distance between X and Y.
My reasoning was, for all $(x, y) ...
1
vote
2answers
141 views
Simplification of the Expected Value via CDF: Does it work for ALL Probability Distributions?
If a random variable $X$ has a density $f$, then the expected value can be simplified:
$$\mathbb{E}[X]=a+∫_{a}^{b}(1-F(x))dx,$$
where $F$ is the cumulative distribution function, $F(x)=\Pr(X≤x)$.
...
1
vote
1answer
410 views
Integrating a probability density function
Let the pdf defined as: $P(x, \bar{x}, \sigma) = \exp\left(\frac{-(x-\bar{x})^2}{2\sigma^2}\right)$. How can we integrate this probability density for some values of $x$ that are higher than a given ...
3
votes
2answers
3k views
How can a probability density be greater than one and integrate to one
Wikipedia says:
The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.
and it also says.
Unlike a probability, a probability ...
1
vote
1answer
125 views
Trouble with basic integration
I'm doing the something that results in following integral:
$$f(z) = \int_{-\infty}^\infty \frac{1}{2\pi}x\exp\left(\frac{-x^2}{2}\left(1+z^2\right)\right) dx$$
Then since $f(z)$ is even we get:
...
1
vote
1answer
103 views
Does an inequality between definite integrals imply an inequality between the derivative wrt an exponent?
I have two cdfs, both distributed over 0 to 1.
Let's call them $F(x)$ and $G(x)$.
If I know that
$$\int_0^1 F(x) \,dx < \int_0^1 G(x) \,dx$$
then, does it follow that
$$ \left|\frac{d}{dn} ...
1
vote
2answers
251 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
...
1
vote
0answers
174 views
Verifying Normalization
I have encountered an exercise question asking the reader to verify that a wavefunction is normalized. So I calculated the probability density -- $|{\psi}|^2$, then verified that the integral does ...
