0
votes
1answer
29 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
64 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
45 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
24 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
28 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
17 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
48 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
35 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
16 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
83 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
34 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
23 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
27 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
14 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
18 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
45 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
15 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
36 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
26 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
35 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
27 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
32 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
13 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
28 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. ...
0
votes
0answers
33 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
1
vote
1answer
42 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
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, & ...
1
vote
1answer
56 views

Is the derivative of a continuous density function on $\mathbb{R}$ integrable?

Any continuous density function on $\mathbb{R}$ is integrable. Is it true that all derivatives of a probability density function defined on $\mathbb{R}$ is also integrable?
0
votes
0answers
48 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
33 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
1answer
25 views

Calculating the distribution of the minimum of two exponential functions

Suppose X and Y are two independent exponential random variables with rates $\alpha$ and $\beta$ respectively. I know the following equality to be true but I don't know why it's true: $\mathbb{P}(Y ...
0
votes
1answer
45 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct a probabilistic model, I have tried different methods of approach, by parts, substitution, but to no avail. Any help with this would be greatly appreciated!
0
votes
1answer
13 views

Finding Marginal Density function of a joint density function

$$f(y_1,y2) = (1/2)y_1 + (1/4)y_2,\\ 0 \le y_1 \le 1, \\0 \le y_2 \le 2 \\ 0\text{ elsewhere.} $$ How would I find the marginal density function for $Y_1$, and $Y_2$? To find $Y_1$ you would do ...
1
vote
0answers
48 views

Minimum of N Chi-square random variables when N is large

I have a problem in numerically evaluating the PDF of $Y=\min(X_1,X_2,\cdots,X_N)$ where $N=\binom{M}{K}$, the binomial coefficient and $X_i$s are iid Chi-square random variables. The CDF of $Y$ is ...
1
vote
0answers
35 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 ...
1
vote
1answer
45 views

Probability that there is an edge between two nodes in a random geometric graph

I am new to Random geometric graphs. I have a graph with vertices being generated uniformly over $[0,1]^2$. There is an edge between two vertices if the Euclidean distance between the two vertices is ...
0
votes
0answers
34 views

Proof convolution formula two stochastic variables

Let's say I have two continuous independent stochastic variables, defined on $(0, \infty)$. With densities: $X_1$ ~ $f_1(t_1), t_1 \in (0, \infty)$ $X_2$ ~ $f_2(t_2), t_2 \in (0, \infty)$ The ...
0
votes
1answer
30 views

Lifetime of exponential variable of a battery

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type ...
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
2answers
107 views

What does integration of g(x)f(x)dx mean with known bounds?

What does $\int_{0}^{a}g(x)f(x)dx$ mean if $f(x)$ is the probability density function of $X$? I do know $\int_{-\infty}^{+\infty}xf(x)dx$ is the expected value of $X$, from Wikipedia. But I am not ...
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 $0 \leq x \leq 4$; ...
0
votes
1answer
187 views

Derivative of Expected value

Say we have a random variable $X$, with density function $f(x)$, and moment generating function $M(t) = E[e^{tX}]$, and we take the derivative - $M(t) = \frac{d}{dt}E[e^{tX}] = E[\frac{d}{dt}e^{tX}]$ ...
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 ...
1
vote
1answer
35 views

Expected value of a random variable $X$ with density function $f(x) = \frac{5}{x^2}$ when $x > 5$

I am looking for $E[X]$ when $X$ has a density function $$f(x) = \frac{5}{x^2}$$ when $x > 5$ and $0$ elsewhere. But $\int_5^\infty x \frac{5}{x^2}dx = 5\int_5^\infty x^{-1}dx = ...