# Tagged Questions

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### 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 [ ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $(\ln ... 1answer 34 views ### Weird question about probability density function I'm assuming "actual" means the total probability of the PDF (the integral from$-\infty to \infty$) must be 1, so $$\int\limits_{-\infty}^{\infty} ke^{-0.1t}dt = 1$$ Wolfram Alpha seems to be ... 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 ... 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 ... 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: ... 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 ... 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 ... 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$\sum_{i=1}^4\theta_i= 1$... 0answers 31 views ### Lifetime of pdf disk The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows:$f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ... 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$0 \leq x \leq 4$; ... 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, ... 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): $$F(x) = \int_0^\infty f(x,y)\,dy. ... 1answer 49 views ### Statistics: Integration from a joint probability distribution If the joint probability density of two random variables is given by:$$f(x_1, x_2) = \begin{cases}6e^{-2x_1-3x_2} &\quad \text{for } x_1 > 0,\, x_2 > 0\\ 0,&\quad ... 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 ... 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 ... 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$\Omega$... 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 ... 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. 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 ($1$)$\rightarrow$... 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 ... 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. 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 ... 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 $. I have ... 1answer 49 views ### Joint To Marginal Density : Can't figure it out. Here goes the problem: Problem: Suppose$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$. ... 0answers 87 views ### Adding truncated normals: calculating convolutions Problem: Suppose that$X$,$Y$, and$Z\$ are independent standard normal random variables. What is the probability of: P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, ...
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 ...