0
votes
1answer
46 views

Need an example/counterexample of continuous and increasing function.

If $\mu$ is a finite measure on the measurable space $\big( X, \mathscr{F} \big)$, $f : X\to [ 0, +\infty)$ is measurable. Then $\textbf{does it exist a continuous function $g : [ 0, +\infty)\to [ ...
1
vote
1answer
45 views

How to compute the integrals in inverse formula?

I have following characteristic function for certain random variable X: $$\Phi (t) = \frac{\beta_1\beta_2}{\eta_1}\frac{\eta_1 - it}{(\beta_1 - it)(\beta_2 - it)}$$ where $\eta_1 > 0, \quad\beta_1 ...
0
votes
1answer
39 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% ...
1
vote
0answers
19 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
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 ...
3
votes
2answers
64 views

$\frac{1}{\sqrt{2\pi}}\int_\frac {1}{2}^0\exp(-x^2/2)dx$

How do we analytically evaluate $J=\frac{1}{\sqrt{2\pi}}\int_\frac {-1}{2}^0\exp(-x^2/2)dx$? This is what I tried: $$ J^2=\frac{1}{{2\pi}}\int_\frac {-1}{2}^0\int_\frac {-1}{2}^0\exp(-(x^2+y^2)/2)dxdy ...
0
votes
0answers
15 views

Probability density function of an element

How to find the probability density function of $x_m\left(1\le m\le n\right)$ from joint density function, $p_X\left(x_1,x_2,\cdots,x_n\right)$, of $n$ random variables which satisfy following ...
0
votes
0answers
56 views

Exponential integral with $x^2$ and $\cos x$

The first part is just a Gaussian integral and the second is the modified Bessel function of the first kind for $n=0$, but I couldn't find any information and what to do with their summation. Any tips ...
0
votes
1answer
31 views

A moment's question.

Let G be a (absolutely) continuous distribution such that $$\displaystyle{\int_{-\infty}^{\infty}{x^{2}dG(x)}}<\infty$$ or $$\displaystyle{\int_{0}^{1}{\left[G^{-}(t)\right]^{2}dt}}<\infty.$$ ...
0
votes
1answer
27 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 ...
3
votes
1answer
49 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
2answers
36 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} ...
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?
0
votes
0answers
13 views

How to find the CDF of distance between two point in two circles respectively?

Let $C_1$ and $C_2$ be two circles with radius $R$ and $r$, $H$ be the distance between two centers, $H\in[0,+\infty)$, pick up Point $P_1(x,y)$ from $C_1$ and Point $P_2(a,b)$ from $C_2$ uniformly, I ...
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
1answer
63 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
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
32 views

Joint distribution: show the components of the joint distribution are independent.

Very odd question I think... Show that if $(X,Y)$ is a random vector in $\mathbb{R}^{2}$ with density $f_{(X,Y)}(x,y) = f(x)g(y)$ for a pair of non-negative functions $f$ and $g$, then $X$ has ...
1
vote
1answer
37 views

Resolve integral with importance sample Monte Carlo

I'm trying to compute the integral $$\int_{a}^{b}(\sin( 1 + x ) + \cos( 1 + x ))e^{-x}\ dx$$ using importance sample Monte Carlo method. The exercise ask to use Cauchy Distribution to resolve the ...
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
3answers
46 views

Integrating the product of Poisson and exponential pdf

So I'll spare the background as to why, but I'm trying to integrate the following: $$\int_0^{\infty} \frac{e^{-(\lambda+\mu)t}(\lambda t)^n}{n!} dt$$ If you parameterize a Poisson w/ $\lambda$ and ...
1
vote
1answer
38 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
41 views

the marginal pdf of a transformed variable from a joint distrubution

The questions tells us to let X and Y be random variables for which the joint p.d.f. is as follows: $$f(x,y)= \begin{cases} 2(x+y), & \text{for $0 \le\ y \le\ x \le\ 1$} \\ 0, & ...
0
votes
0answers
49 views

Is $(\ln l(y))^2 l(y)^x f_0(y)$ integrable over the real numbers?

Is $(\ln l(y))^2 l(y)^x f_0(y)$ integrable over $\mathbb{R}$ for any continuous pair of densities $f_0$, $f_1$ and $l=f_1/f_0$ with some known constant $0\leq x\leq 1$? It seems that $(\ln ...
1
vote
1answer
34 views

Weird question about probability density function

I'm assuming "actual" means the total probability of the PDF (the integral from $-\infty to \infty$) must be 1, so $$\int\limits_{-\infty}^{\infty} ke^{-0.1t}dt = 1$$ Wolfram Alpha seems to be ...
0
votes
2answers
58 views

Difficulty finding Expectation of a special function

I have a special function given as: $${\rm f}\left(r\right) ={1 \over \beta\lambda}\,2^{r/\beta} \exp\left({\left[2^{r/\beta} - 1\right]K \over \lambda}\right)$$ I should find the Expectation of ...
1
vote
0answers
36 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
5
votes
6answers
358 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: ...
0
votes
1answer
28 views

Integration of a multiplied weibull distribution

I am having trouble integrating a weibull function multiplied with a production function. The expression shortened for integration is as follows: $$v^{k+2}e^{-\left(\frac{v}{y}\right)^k}$$ I hope ...
0
votes
1answer
33 views

Scaling the Lebesgue-Stieltjes integral

Suppose that $F$ is a distribution function. Denote by $\mu_F$ the measure on $\mathbb{R}$ induced by $F$. Suppose that $a>0$. Define a new distribution function $F_a$ by $F_a(x):= F(ax)$, and ...
0
votes
0answers
15 views

Integration over multinomial model paramters

I come across this integral over four variables $(\theta_1, \theta_2, \theta_3, \theta_4)$, which are multinomial distribution parameters, in which $k=4$, so $\sum_{i=1}^4\theta_i= 1$ ...
1
vote
0answers
31 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
1
vote
1answer
31 views

Random variable of a store

The weekly profit in thousands of dollars of Miller's Office Supply Store is random variable X whose cdf is given as follows: $F(x)=0$ for $x<0$; $F(x)=(3/32)(2x^2-x^3/3)$ for $0 \leq x \leq 4$; ...
1
vote
0answers
16 views

Gram-Charlier expansion, option price, higher derivative and integration by parts

I am currently reading a finance paper of Backus et al. (2004), called 'Accounting for biases in Black-Scholes'. To explain an abnormality called 'volatility smirk' that can be found in option prices, ...
4
votes
2answers
97 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
49 views

Statistics: Integration from a joint probability distribution

If the joint probability density of two random variables is given by: $$f(x_1, x_2) = \begin{cases}6e^{-2x_1-3x_2} &\quad \text{for } x_1 > 0,\, x_2 > 0\\ 0,&\quad ...
0
votes
1answer
29 views

Appreciate help with solving a probability density function for its constant term

I am using StackOverflow a lot for asking and answering programming related questions, and I hope it is appropriate if I'd ask my question below on here on this sister-site. If not, please let me know ...
0
votes
1answer
32 views

I have some approximate integral calculation. Is there a clean way to prove it?

Let: $P(R)=\int_R^{\infty}F(z)e^{-z}dz$ where $F(z)$ is the CDF of some discreate positive R.V. denote by $U$. Integrate by parts: $P(R)=(-F(z)e^{-z})_R^{\infty}+\int_R^{\infty}F'(z)e^{-z}dz$ The ...
-1
votes
1answer
48 views

Constructing a weighting function with equal mean on two random variables

I am not a mathematician, but I hope that it is understandable. I try to tackle a problem which can be described as the following: Let $X_1$ and $X_2$ be random variables with same support $\Omega$ ...
2
votes
0answers
96 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
2
votes
2answers
83 views

how to solve this integral in survival analysis

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
0
votes
0answers
161 views

Is the summation of given $3$ integrals always greater than $1$

For two density functions $f_1$ and $f_0$ on $\mathbb{R}$, $l(y)=f_1/f_0(y)$ is an increasing function of $y$. We are also given the following information: Condition ($1$) $\rightarrow$ ...
2
votes
1answer
146 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
1
vote
2answers
42 views

Calculate the value of c for which f is a probability density.

Let f the function defined by: Where c is positive none zero and constant . How can i calculate the value of c for which f is a probability density.Thnxs for the help.
3
votes
1answer
108 views

How to arrive at a specific formulation of the relative median deviation? Related to integration and statistics.

So my title is not very specific but here is the question in more detail. I am an economist currently working with this book: Frank Cowell - Measuring Inequality On page 25 a formulation of the ...
5
votes
5answers
120 views

Please explain to me why the Expected Value is $ E[X] = \int_{-\infty}^{\infty} x f_X(x) dx $

For probability density functions (at least for the normal distribution and beta distribution) it holds that the expected value is given by $ E[X] = \int_{-\infty}^{\infty} x f_X(x) \, dx $. I have ...
1
vote
1answer
49 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 $c$. ...
1
vote
0answers
87 views

Adding truncated normals: calculating convolutions

Problem: Suppose that $X$, $Y$, and $Z$ are independent standard normal random variables. What is the probability of: \begin{equation} P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, ...
1
vote
1answer
99 views

On the mgf of the Logistic Distribution

So the Logistic Distribution pdf (w/ mean = 0 and shape parameter = 1) looks like this: $$f_X(x)=\frac{e^{x}}{({1+e^{x}})^2}\;\;, \;\;-\infty<x<\infty$$ Now, I am interested in getting its ...