# Tagged Questions

2answers
50 views

### Expected value of ordered statistics

Random vector $(X,Y)\sim N(0,0,1,1,\rho)$, that is to say, the density function of $(X,Y)$ is given by f(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left\{-\frac{1}{2(1-\rho^2)}(x^2-2\rho ...
1answer
31 views

### Probability of line intersecting the convex set.

I would like to prove this theorem: Let $A,B \subseteq \mathbb{R} ^3$ be convex, limited sets. $B \subseteq A$. I have a "random line", which intersects A. Probability, that this line also intersects ...
1answer
74 views

### Finding the median value on a probability density function

Quick question here that I cannot find in my textbook or online. I have a probability density function as follows: $$0.04x \space 0 \le x < 5$$ $$0.4 - 0.04x \space 5 \le x < 10$$  0 ...
1answer
46 views

### Independence between Uniform distribution and Exponential distribution question

I am trying to solve the following problem and I am having a great deal of difficulty in a number of areas. Help would be greatly appreciated! Let me state the problem first. If $X$ is uniformly ...
1answer
34 views

### Limits of Integration for marginal pdf

I just had a small question as something is bothering me. I am trying to find the marginal pdf of the following joint pdf: $f(x,y) = (1/8)(y^2 - x^2)e^{-y}$ where $-y \le x \le y$, \$0 < y < ...
0answers
70 views

### Tight Upper/Lower bound for Incomplete Gamma function

Does anyone know of any tight upper/lower bound for incomplete Gamma functions? i.e either of the following functions: $$\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t$$ or  \gamma(s,x) ...
0answers
81 views

### Bayesian posterior with integrals over normal densities

Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
0answers
33 views

### Average Bhattacharyya distance

Two conditional PDF's $f_1(y|X=x)\sim N(Ax,\sigma^2)$ and $f_2(y|X=x)\sim N(-Ax,\sigma^2)$ are given. I need to find their average Bhattacharyya parameter assuming $X$ has a Rayleigh distribution ...
1answer
59 views

### Calculate expected values for a normal distribution

Suppose that X ~ N(1,2) Find: $E(X-1)^4 and E(X^4)$ I have no idea how to get started. Do I just need to integrate the pdf of the normal distribution multiplied by what is between the brackets? If ...
1answer
29 views

### Find the moment generating function

$$f(y)=\frac{e^{-|y|}}{2}$$ I tried calculating it by integrating $(1/2)e^{ty}\cdot e^{-|y|}) dy$ and splitting up that integral into two separate integrals. However, I did not get a finite answer. ...
1answer
41 views

### Evaluate $Pr[Y - X < c, X \ge 0]$

Let $f(x,y) = Pr[X=x, Y=y]$ be the joint density of two random variables, $X$ and $Y$. I have been given s = $Pr[Y - X < c, X \ge 0] = \int_0^\infty \int_{-\infty}^{c+y} f(x,y) dx dy$ I am a ...
1answer
120 views

### Probability: Determining Which Phone Plan Is Better

A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of $10$ cents per minute, whereas the second charges a flat rate of $99$ cents for calls up ...
0answers
47 views

### Expected value of threshold function

Let $X$ be a random variable with probability density function $f(x)$, $\alpha>0$ a constant and $g(x)$ a function with \$g(x)=\begin{cases} \begin{array}{c} g_{0},\\ g_{1} \end{array} & ...
1answer
40 views

### Finding $EX$ of a density function (integrating $\ln u$ over infinity)

I've been given a density function as: $f(x) = 1/4e^{-|x|/2}$ where $-\infty < x < \infty$ and need to show that $EX = 0$ I understand that to find $EX$ I must calculate $\int xf(x)~dx$ ...
1answer
89 views

### Erroneous Answer Key?

The problem I am working on is: The current in a certain circuit as measured by an ammeter is a continuous random variable $X$ with the following density function: $f(x) = .075x + .2$ for \$3 \le x ...
1answer
167 views

### Taking the derivative of definite integral?

I'm having trouble understanding the derivative of definite integral. For example, why is the following true? \$\frac{d}{dx}\displaystyle\int_{0}^{x}F_{1}(x-v)f_{1}(v)\, \mathrm{d}v = ...
1answer
82 views

### How can I solve this integral?

How can I solve the following integral? \int_{-\infty}^\infty \prod_{i=1}^n \bigg( 1 - \Phi\left(\frac{c - \mu_i}{\sigma_i}\right) \bigg) \frac{1}{\sigma_Y}\phi \bigg(\frac{c-\mu_Y}{\sigma_Y} ...
1answer
60 views

### Integrate Gaussian PDF over another gaussian, to get probability of imprecise datapoint

I have a Gaussian PDF and an observation datapoint. To get a probability, I cannot use the datapoint itself--I need a range around it. But say the datapoint is imprecise and obeys a normal ...
0answers
75 views

### Probability involving Method of Moments

Considering a box containing balls labeled 1 through $\theta$ from which we sample $n$ balls (hence, a discrete uniform distribution on the interval of $(0,\theta),$ so the probability of picking ...
3answers
136 views

### Binomial distribution with Uniform parameter

I have a problem with following exercise (it comes from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001, page 155, ex. 6): Let $X$ have the ...
1answer
71 views

### Techniques for evaluating probability integral

Consider the integral of a normal distribution: $$\int_a^b f(x)\,\mathrm d x=c$$ and a second integral for the expected value: $$\int_a^b x\cdot f(x)\,\mathrm dx$$ Since you know the first ...
1answer
63 views

### Expected value where benefit and probability depends on stochastic variable

I am trying to calculate the expected benefit of an action, where both the benefit and probability of carrying out the action depends on a parameter c, which is distributed according to f, F'=f. The ...
3answers
248 views

### Compute probability of a particular ordering of normal random variables

There are $m$ normally distributed, independent random variables $N_1, \ldots, N_m$ with distinct means $\mu_1, \ldots \mu_m$ and standard deviations $\sigma_1, \ldots, \sigma_m$. Then, we get a ...
3answers
31 views

### A bayesian way of calculating the probability $\text{Pr}(\theta \in (\theta_1, \theta_2) | y )$

I'm confused by a bayesian way of calculating the probability $\text{Pr}(\theta \in (\theta_1, \theta_2) | y )$, where $\theta$ is assumed to have a (prior) uniform distribution on $[0,1]$, and the ...
1answer
47 views

### evaluating expected values or integrals

When $x$ and $y$ are two iid random variables, I want show if $$E(x\mid x>y)<1$$ can be determined: [1] without knowledge of the distribution of the random variables; and [2] if $x$ and $y$ ...
0answers
75 views

### conditional expectation — simplify

Let $x$ and $y$ be two independent random random variables with densities $f(x)$ and $f(y)$. I intend to define $\int \int_{-\infty}^x f(x)f(y)\,dy\,dx$ in relation to $E[x\mid x>y]$. I attempted ...
2answers
131 views

### Finding PDF involving absolute value

I'm trying to solve the following question: Given an exponential R.V. X with rate parameter $\lambda > 0$, find the PDF of $V=|X-\lambda|$. In order to find the PDF, I would like to use the CDF ...
2answers
165 views

### Find $P(X/2 < Y < X)$

Hello people I have a small confusion regarding a given problem. Lets see it a bit. We have given that: \$f(x,y) =\begin{cases} x y \text{ if } 0 \leq x \leq 2 \text{ and } 0\leq y\leq 1 \text{ ...
2answers
56 views

### Expectation and Distribution Function?

Consider X as a random variable with distribution function $F(x)$. Also assume that $|E(x)| < \infty$. the goal is to show that for any constant $c$, we have: \int_{-\infty}^{\infty} x (F(x + c) ...
0answers
71 views

### second order stochastic dominance

Let the nonnegative random variables $X$ and $Y$ have distribution functions $F$ and $G$ and density functions $f$ and $g$, respectively. Suppose $X$ is second-order stochastically dominant over $Y$, ...
1answer
65 views

### Upper bounds for an integral with an infinite upper limit

I'm trying to work out an upper bound for the following problem, but I'm making very little progress. Hopefully, someone will be able to make a suggestion. The integral I'm attempting to bound is: ...
1answer
86 views

### Expectation of a multivariate Gaussian over a plane

For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation : $E[X|X^Tb = c]$ ...
1answer
67 views

### Evaluating $\int_{0}^{1}\int_{0}^{1}\frac{r^{i+j}(1-r)^{k+l}s^{2m-i-j}(1-s)^{2m-k-l}}{(r+s)^{m}(2-r-s)^{m}}drds$

I'm trying to compute a closed form expression for the integral  \int_{0}^{1}\int_{0}^{1}\frac{r^{i+j}(1-r)^{k+l}s^{2m-i-j}(1-s)^{2m-k-l}}{(r+s)^{m}(2-r-s)^{m}}drds \quad i,j,k,l ...
0answers
109 views

### Change of variables problem

I came across this problem yesterday where i wanted to change variables in an integral like below. $$\iiint f\left(x_{1},x_{2},x_{3}\right) dx_{1}dx_{2}dx_{3}\tag{1}$$ so $y_1 = x_2 - x_1$ and \$y_2 = ...
2answers
205 views

### Expected value of applying the sigmoid function to a normal distribution

Short version: I would like to calculate the expected value if you apply the sigmoid function $\frac{1}{1+e^{-x}}$ to a normal distribution with expected value $\mu$ and standard deviation $\sigma$. ...
2answers
94 views

### Determining a value of $c$ such that $x/c$ for any $x$ equals $1/2$

$x$ is a variable which take its real values in the interval $[\min x, \max x]$ and $c$ is a real constant value that I want to determine. I want to determine a fixed value for $c$ such that $x/c$ ...
2answers
95 views

### Evaluation of probability related integral

I have encountered the following integral in my research which does not give-in to my attempts:  \int_\mathbb{R} x \left( \frac{1}{\sigma_1} \phi\left(\frac{x}{\sigma_1}\right) ...
0answers
107 views

### Derivative of Riemann integral [of  a distribution function].

I am having a trouble solving a derivative of a Riemann integral while trying to obtain a distribution of a variable being a function of another random variable. Let $X$ be a random variable with ...
4answers
815 views

### Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
1answer
54 views

### Integral of unimodal functions $f>0$.

Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is positive almost everywhere and integrable. We know that if $n=1$ and if $f$ is unimodal then the integral $F(x)=\int_{[-\infty,x]} f$ is convex for ...
1answer
102 views

### Expectations containing normal CDF

Suppose that $X\sim\mathcal{N}\left(0,1\right)$ (i.e., $X$ is a standard normal random variable) and $a,b,$ and $c$ are some real constant. Does any of the following expectations have a closed-form? ...
1answer
105 views

### How demonstrate the Craig representation for the Gaussian probability function?

The Q-function is defined by : $$Q(x) =\frac{1}{\sqrt{2\pi}} \int_{x}^{\infty}\exp(-\frac{u^2}{2}) \ \mathrm{d}u \ \ (1).$$ According to the wiki page there is an alternative form of the Q-function ...
2answers
368 views

### How to compute this integral involving a cdf?

$\int_0^\infty\Phi(\frac{-x}{\sqrt{2}})d\Phi(x)=?$ where $\Phi(x)$ is the cumulative distribution function of a standard normal random variable.
1answer
131 views

### Probability that a Multivariate Normal RV lies within a Spherical Region of Radius R

I am currently using different procedures to estimate the probability that a $D$-dimensional Gaussian random variable with mean $\mu$ and covariance $\Sigma$ lies within a sphere of radius $R$ that ...
1answer
112 views

### The probability of $Ax^2+Bxy+Cy^2 = 1$ defining an ellipse.

In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian ...
0answers
126 views

### a integral of bivariate Gaussian random variables.

I met the following problem when doing estimation and detection homework. The problem asks for a maximum likelihood estimator for (v,$\rho$) of bivariate joint Gaussian, where v is the common ...
2answers
231 views

### Integral of probability density function

$$\int_{-1}^{1}{\int_{x^2}^{1}{cx^2y \ dy \ dx}}$$ Given that: $\int_{-1}^{1}{\int_{x^2}^{1}{cx^2y \ dy \ dx}}=1$, I have to find $c$. I am a bit disoriented by that $x^2$ as the limit of the first ...
2answers
142 views

### How can I prove $\int[F(x+a)-F(x)]\,dx=a$

How can I prove $$\int[F(x+a)-F(x)]\,dx=a$$ where $F(x)$ is the cumulative distribution function?
3answers
546 views

### What is the insight behind the Lebesgue integral?

Edit 3: OK, I had an insight, inspired in part by Ben-Blum Smith's comment, and the post he linked to. (I have no idea if this insight is right; it's barely a hunch, and that's why I'm not submitting ...
1answer
127 views

### how to code PDF ? or for that matter Integration in general.

I need to use the Probability Function on a set of values. I am okay with understanding the math, but i don't quite know how to convert that stuff to code. basically i need the integral of ...
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