1
vote
1answer
36 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
34 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
46 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
30 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
54 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
29 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
304 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
22 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
26 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
6 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
30 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
9 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
81 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
40 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
27 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 ...
0
votes
0answers
19 views

Maximize integral of product of gaussian

Consider the Gaussian density function $f: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ defined as $$ f( x, \mu, \Sigma ) := \frac{1}{ \sqrt{(2 \pi)^n ...
-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
83 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
75 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
158 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
79 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
38 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
106 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
107 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
42 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
59 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
53 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 ...
1
vote
1answer
48 views

How do you find the distribution of this sum?

If $X\sim \text{Normal}(\mu=0, \sigma^2), Y\sim \text{Unif}(0,\pi)$, and $X \perp Y$, how do you find the distribution of $Z=X+a\cdot cos(Y)$ for some $a > 0$ ? I've found the distribution ...
0
votes
1answer
67 views

Finding integration bounds for density of sum of two independent random variables

Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a < 1$} \\ 0 & ...
1
vote
2answers
92 views

Integrating a special skew normal — the CDF of a convolution of a normal with a truncated normal

I am having a little trouble trying to compute an integral. In short, I wish to solve the following: $$F(x) = \int_{-\infty}^x \phi(au-b)\,\Phi(au+b)\,du $$ My intuition is that this might be ...
1
vote
1answer
47 views

The intergral $I=\int _0^{\beta }f(x)dx$ is given,for $\alpha,\beta \in \mathbb{Z}$ ,how can we find $\int_0^{\alpha\beta}f(x)dx$ in terms of $I$

i am working with a gaussian normal distribution function in probability,i am given values for the integral when $z\le 4$ and i want to find a value $z=8$,in general if $z=4$ is given , how to find ...
0
votes
0answers
38 views

PDFs with a specific Coefficient of Variation ($\sigma/\mu$)

So, I am trying to find the general expression for the probability density functions that have a specific Coefficient of Variation. The Coefficient of Variation is the ratio between $\sigma$ (the ...
0
votes
0answers
77 views

Average of an exponential over Dirichlet probability distribution on the (n-1)-simplex

Any idea how I can solve this integral for arbitrary integer $n$ ($n \geq 2$) with real non-negative coefficients $\{s_i\}_{i=1,..,n}$: $I(s_1,...,s_n):= \int_0^1 dx_1 ...\int_0^1 dx_{n}\,\, e^{ ...
1
vote
2answers
83 views

independent chi squares mean independent non central chi square?

Let $Y$ be a multivariate normal random vector with covariance $\Sigma$. Let $A_0,A_1$ be matrices such that $$A_0\Sigma A_1=0.$$ It is known that in this case $Y'A_0Y$ and $Y'A_1Y$ are independent ...
1
vote
0answers
22 views

Anti-derivative of a function involving exponentially distributed variable

Suppose a random variable $x$ with p.d.f $f(x) = \lambda e^{-\lambda x}$ such that $\lambda$ is the parameter of $f$. Given a function $ g(x) = (a + bx )e^{- \frac{\lambda x}{a}} $ where $a,b \in ...
0
votes
1answer
73 views

Limits of integration for a joint PDF

I have $f_{X,Y}(x,y) = \lambda^2e^{-\lambda y}$ for 0 < x < y. If I want to show that this is a joint PDF, I need to do a double integral and show that it is equal to 1. Do I set my integration ...
1
vote
0answers
89 views

Simplify the expectation of the maximum of two random variables

My aim is to simplify the maximum of two expressions each of which are a function of exponentially distributed random variables Given: positive constants $a,b,c,d$. Independent random variables $x,y ...
0
votes
0answers
47 views

Integrating Log-Logistic PDF times x and ln(x)

I am attempting to integrate the log-logisitc PDF multiplied by x and ln(x) on an interval from a to b, in particular (using the parameters from wikipedia): $$\int_a^b ...
1
vote
1answer
72 views

Variance of this probability density

I have the function $\rho(x) = \frac{sin^2(x)}{x^2}$ and I want to calculate its variance on $\mathbb{R}$. Does anybody know how to do this? Cause afaik the integral does not converge.
1
vote
1answer
128 views

Find cumulative distribution function of a continuous random variable.

$X$ is a random variable with density $f(x)=0.5e^{-|x|}, (-\infty<x<\infty)$. Find c.d.f of $x^2$. I dont quite get the hang of these. I tried for just x and got the following. for $x<0$: ...
1
vote
2answers
110 views

Integrating exponential of exponential function

I would like to find the integral of $\int_0^\infty\exp(-u-\exp(-ku))\,du$ for $k>0$. This is related to the gumbel distribution(http://en.wikipedia.org/wiki/Gumbel_distribution), which shows ...
0
votes
2answers
50 views

About probability density function

I have a question about probability density function in my book. It reads: A Probability density function is of the form $p(x) = Ke^{-a|x|}$ , $x \in (-\infty,\infty)$.The value of $K$ is: 1] ...
0
votes
2answers
32 views

Finding the mass generating function of a continuous random variable given a pdf

The pdf is given, $$f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$$ Where $x\in(-\infty,\infty)$, $\sigma>0$, $\mu\in(-\infty,\infty)$. ...
0
votes
2answers
95 views

Joint distribution of U = X + Y and V = X - Y

I have two independent continuous random variables, X and Y, which are uniformly distributed over the interval [0,1]. From this I have two further random variables, U and V, which are defined as U = X ...
-1
votes
2answers
74 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
0
votes
1answer
33 views

Computation of an integral

While computing densities for some distributions, I stumbled on the following family of parametrized integrals: $$ p (x) := \sqrt{\frac{2}{\pi}} \int_{\mathbb{R}_+} e^{-\frac{x^2}{2 y^2} - y^2} \ d ...
2
votes
1answer
71 views

A variance-mixture model

So I've tried to make a probability distribution which has a tunable degree of kurtosis and which becomes Gaussian if the control-parameter goes to 0. Now Levy-distributions are out of the question, ...
2
votes
1answer
57 views

How to finish some complex integration

How to finish some integration as following below: $$\int_x^{\infty} \frac{\mathrm \beta^{\alpha+\gamma} X^{\alpha-1}(y-x)^{\gamma-1}\exp^{-\beta y}}{\Gamma(\alpha) \Gamma(\gamma)}dy\;$$ and ...