0
votes
0answers
20 views

Integral with a gamma functions inside

I have a function based on the binomial distribution, $$f(x;n,p)=\sum_{i=0}^{n} |x-i|\binom{n}{i} p^i (1-p)^{n-i}.$$ It's not so hard to plot this out with discrete points, but I'd like to smooth ...
0
votes
0answers
272 views

Inverting a Characteristic Function for half-cubic Student T entailing a Modified Bessel of 2nd kind

The Characteristic function of the Student T with $\alpha$ degrees of freedom, $C(t)=\frac{2^{1-\frac{\alpha }{2}} \alpha ^{\alpha /4} \left| t\right| ^{\alpha /2} K_{\frac{\alpha ...
1
vote
0answers
25 views

derivative of normalizer in exponential form — change integral and gradient

When deriving the relation between normalizer and expectation of the sufficient statistic for distributions in exponential form one uses the fact, that the density integrates to one: $$1 = ...
0
votes
2answers
46 views

Where did I go wrong with this integration?

Calculate $E\!\left[X^2\right]$ with $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ with $x \in \mathbb{R}$. Well $f_X$ is symmetric about $0$ so this is the same as \begin{equation*} ...
1
vote
1answer
101 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
1
vote
0answers
49 views

Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
0
votes
2answers
113 views

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 ...
0
votes
5answers
90 views

How to prove that the function $f(x)=0.1\,e^{-0.2|x|} $ is a probability density, and then use it?

So here's the integral, I'm having a hard time solving it. I even tried integration software, but it didn't help: $$ I=\int_{-\infty}^{+\infty}f(x)\,dx,\qquad f(x)=0.1\,e^{-0.2|x|} $$ The question ...
1
vote
1answer
21 views

Cumulative probability of Chi-squared distribution

If $X$ is distributed $\frac{\chi_{10}^2}{10}$ , find the probability that $X > 1.83$ The formula for the Chi-squared CDF I'm using is the following, which is the integral of the PDF formula: ...
0
votes
1answer
22 views

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 ...
0
votes
1answer
40 views

Gamma and a poisson distribution

All I know is that $P(X=x)=e^{-5x}\frac{5^x}{x!}$ where $x\geq 0$. The formula that im using is $E(Y)=\int E(Y|X=x)f_x(x) dx $ where $f_x(s)\int f(x,y) dy $=. Also I guess that from the hint that ...
0
votes
1answer
36 views

Calculate the expected value

To get the expected value of $E(X), E(Y) $ and $E(X, Y)$ given: $$ f_{X,Y}(x,y) = 3x $$ where $0\le x \le y \le 1.$ My solution is, first get the margin distribution: \begin{aligned} f_x(x) &= ...
0
votes
1answer
13 views

Continuous probability for time overlap

The following is an example from Introduction to Probability by Dimitri P. Bertsekas example 1.5. I am having trouble deriving the answer for the example X and Y have a meeting at a given time and ...
0
votes
2answers
37 views

how to prove the expectation of 1/x^2 when x is standard normal does not exist?

I am trying to show that the expectation of $\frac{1}{X^{2}}$ does not exist if $X$ is standard normal, but do not know how....could anyone help, please? The integration is $\int_{-\infty}^{\infty} ...
0
votes
1answer
46 views

Integrating probabilities

My following problem is of general nature, here is an example to illustrate it. For example let $\left(\xi_i\right)_{i \geq 1}$ be independent and identically Exp(1) distributed random variables. We ...
0
votes
1answer
29 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
1
vote
2answers
48 views

Integration by parts

Integrate using integration by parts: $F(y) = (y+1)e^{-y}$ Find: Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts. Thus far, I've distributed the $e^y$ term and split ...
0
votes
2answers
73 views

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 ...
0
votes
0answers
22 views

Is the following limit of an integral positive?

In one of my problems I need to show if the following holds $$0<\lim_{{g\rightarrow f},\,{\alpha\rightarrow\infty}}\int_\mathbb{R}\log^2(g/f)(g/f)^\alpha f \, \mathrm{d}\mu<\infty$$ Here $g$ ...
0
votes
1answer
47 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
1
vote
0answers
31 views

Integral involving complimentary error function.

Essentially, I am trying to work out an integral of the form: $\int_{0}^{\pi }{x\cdot a\cdot \cos \left( x \right)e^{-\left[ a\cdot \sin \left( x \right) \right]^{2}}\mbox{erfc}\left[ -a\cdot \cos ...
0
votes
2answers
24 views

Marginal density function question

The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, ...
0
votes
1answer
42 views

Determine the target weight so that no more than 5% of boxes with normal weight distribution contain less than 500 g [closed]

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. Suppose a law states that no more than 5% ...
0
votes
2answers
34 views

Probability that the call will be answered at time $t$ is given by $f(t)$. Find the median waiting time for the call.

$$f(t) = \begin{cases} 0 & \text{if $t < 0$ } \\ 0.2e^{-t/5} & \text{if $t\geq 0$} \end{cases}$$. $ $ Find the median waiting time for the call. $ $ I cannot understand ...
1
vote
2answers
47 views

Asymptotic conditional distribution of normal variable

$X$ is a normal variable $\mathcal{N}(0,1)$, $Y$ is a normal variable $\mathcal{N}(n,n-1)$, independent of $X$. I want to prove that the distribution of $X$ conditionally on $X > Y$ is ...
0
votes
1answer
23 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 ...
1
vote
0answers
25 views

How to compute cumulative intensity process integral?

I am faced with a basic question about counting process and its intensity process used in survival analysis. It is actually the textbook example from Aalen's Survival and Event history analysis book. ...
0
votes
1answer
33 views

A moment's question.

Let G be a (absolutely) continuous distribution such that $$\displaystyle{\int_{-\infty}^{\infty}{x^{2}dG(x)}}<\infty$$ or $$\displaystyle{\int_{0}^{1}{\left[G^{-}(t)\right]^{2}dt}}<\infty.$$ ...
2
votes
1answer
81 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
0
votes
0answers
50 views

How to integrate gamma function

I have an exercise with probability and I have troubles with integration of $$ \int_0^t \frac{\lambda^k\cdot x^{k-1}\cdot e^{-\lambda \cdot x}}{\varGamma(k)}dx $$
0
votes
1answer
35 views

Using the Weibull Distribution, derive $E(X^k)$

If $X$~WEI$(\theta,\beta)$, derive $E(X^k)$ assuming $k\gt-\beta$. Note that $X$~WEI$(\theta,\beta)=\frac{\beta}{\theta^{\beta}}x^{\beta -1}e^{-({x}/{\theta})^{\beta}}$ I am having a very difficult ...
0
votes
1answer
31 views

Computing $\int\limits_{p}^{1}\Phi^{-1}(u)\text{ d}u$, $p \in [0, 1]$.

I need to show that $$\int\limits_{p}^{1}\Phi^{-1}(u)\text{ d}u = \phi\left[\Phi^{-1}(p)\right]\text{,}$$ where $\phi$ is the PDF of the standard normal distribution and $\Phi$ is the CDF of the ...
0
votes
1answer
18 views

find marginal density of $X$ where $X,Y$ have joint density $f(x,y)=c\cdot \exp (-(2x+3y))$ over the region $x>0$ and $x<y$.

Find marginal density of $X$ where $X,Y$ have joint density $f(x,y)=c\cdot \exp (-(2x+3y))$ over the region $x>0$ and $x<y$. I've found that $c=15$ for the joint density to be normalized. Then ...
3
votes
1answer
50 views

Interesting Problem - Computing CDF

A rv X is an exponential distribution with parameter 1 and Y is a uniform distribution between 0 and 1. X and Y are independent. Define Z = min {X, Y}. Compute the CDF of Z ? I really have no idea ...
0
votes
0answers
36 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
0
votes
1answer
20 views

Comparing Chebychev's inequality to the exact probability

Let $X$ be continuous with pdf $f(x)=e^{-x}$ if $0<x<\infty$, and zero elsewhere. $(1)$ Use Chebychev's inequality to obtain a lower bound on $P(-1.5<x<3.5)$ Here's what I did: ...
4
votes
4answers
99 views

How to integrate: $\int_{0}^{\infty}e^{tx}(x^2e^{-x})/2dx$

I'm working on a few moment generating function problems and I came across: $f(x)=(x^2e^{-x})/2$ for $x>0$, and zero otherwise. Find the mean. The mean is the same as the expected value. If we ...
0
votes
2answers
37 views

How was this integral set up to compute $Pr(X+Y) \geq\frac{\pi}{2}$?

I am trying to understand how to deal with the following type of question given two random variables $X$ and $Y$ that are jointly continuous with some pdf: Here: $f_{X,Y}(x,y) = \left\{ \begin{align} ...
0
votes
0answers
26 views

Lost a factor while integrating by substitution

In the article this article by Rusell May on a generalisation of the coupon collector's problem the integral (5) is simplified in a special case, resulting in the equation (7). At this point, $L$ is ...
0
votes
1answer
32 views

Integration with respect to conditional measure?

Let $(X_n)$ be a Markov chain. For $i\in S$ my text defines $$N_i:=\sum_{n=0}^\infty \mathbf 1_{\{ X_n=i \}}$$ and then, as a part of a larger proof, claims that $$\mathbb E_i(N_i)=\sum_{n=0}^\infty ...
1
vote
0answers
16 views

Mixture Models: finding the marginal distibution

I don't understand the part underlined in green for b). $f\left( x|Y=y \right)=ye^{-yx}$ but for this to be positive x does not have to be greater than zero?
1
vote
1answer
21 views

Understanding claim in Newman and Barkema's Monte Carlo book

In Newman and Barkema's Monte Carlo Methods in mathematical physics, on page 23-24, the following claim is made: "Assume we have a function f(x) and the integral $I(x)=\int_0^xf(x')dx'$. Then pick a ...
1
vote
0answers
40 views

pdf of area of a circle

$X,Y$ are random variables with standard normal distribution (they are independent). $W$ is the area of the circle that has center at $(0,0)$ and passes through $(X,Y)$. What is the pdf of $W$? I ...
0
votes
2answers
75 views

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: ...
1
vote
2answers
27 views

Joint probability integral

Why is it that: $\displaystyle \int_{-\infty}^{y} f_{X,Y} (u,v) \, dv$ is a function of u?
1
vote
1answer
47 views

In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$

Markov inequality, $$\Pr(X \ge a) \le \frac{E[x]}{a}$$ Proof $$\begin{aligned} E(X) &= \int_0^\infty xp(x)\,dx = \int_0^a xp(x)\,dx + \int_a^\infty xp(x)\,dx \\ &\ge ...
0
votes
1answer
26 views

Integral / Gamma Expectation

I would like to solve the following integral, $\int_{0}^{\infty}\frac{\phi}{a+b\phi} \phi^{c-1}e^{-d\phi}d\phi$. Note $\phi \sim Ga(c,d)$ is a gamma distributed random variable and the integral can ...
0
votes
0answers
18 views

Marginal change in the expected value

Given the function $$ \mathbb{E}[u(t,s,x)]=\int_{X}\max_{t}\left\{ \int_{a}^{b}u(t,s)f(s\mid x)ds\right\} f(x)dx$$ where $f()$ is the pdf, and $X$ is the sample space of $x$. How can i analyze ...
0
votes
0answers
46 views

moment generating function of a shot noise process

A general setting shot noise process $X(\tau)=\sum \limits_k Z(\tau, T_k)$ where $T_k$ Poisson process with intensity $\lambda(t)$, $Z(.,t)$ independent stochastic processes. Show that the moment ...
1
vote
1answer
31 views

Solving a general integral (expectation of some variant of exponential distribution)

Suppose $X$ is distributed exponentially with parameter $\lambda$. Its pdf is $\lambda e^{-\lambda x}$, and the calculation of its expectation is straight forward: $\mathbb{E}(X) = \int_0^\infty ...