0
votes
2answers
39 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
86 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
19 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
35 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
28 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
47 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
71 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
21 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
43 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
23 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
40 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
32 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
46 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
22 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
24 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
80 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
48 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
98 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
20 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
74 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 ...
1
vote
1answer
40 views

derivative of expected value of maximum of two stochastics variables (iid)

I need to optimize an expected value of a maximum value for $q$. The problem has three variables, $q$ is a constant and $D_1$ and $D_2$ are stochastic variables with pdf $f(x)$ and cdf $F(x)$. The ...
0
votes
0answers
40 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
0
votes
0answers
28 views

Probability density function expected values given a cumulative distribution function

I have this probability problem that reads: suppose $f(x)$ is the probability density function of $X$ where $f(x)=0$ unless the values of $x$ either are or in between b and zero ($0 \leq x \leq ...
1
vote
1answer
34 views

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 ...
0
votes
0answers
17 views

Implication of Monotone Likelihood Ratio Property

I can't figure out how a paper is using the monotone likelihood ratio property (MLRP). Here is the statement of the MLRP: The conditional distribution $G(\sigma|c)$ has positive density ...
1
vote
0answers
41 views

Choosing the integral limits for marginal distribution

I've been trying to understand the following: The distribution of two continuous random variables is given by $$f_{X,Y}(x,y)=\frac{3}{7}x\space\space 1\le x\le 2,0\le y\le x$$and $0$ otherwise. ...