Tagged Questions
1
vote
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
$$f(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left\{-\frac{1}{2(1-\rho^2)}(x^2-2\rho ...
1
vote
1answer
31 views
Probability of line intersecting the convex set.
I would like to prove this theorem:
Let $A,B \subseteq \mathbb{R} ^3$ be convex, limited sets. $B \subseteq A$. I have a "random line", which intersects A. Probability, that this line also intersects ...
0
votes
1answer
74 views
Finding the median value on a probability density function
Quick question here that I cannot find in my textbook or online.
I have a probability density function as follows:
$$0.04x \space 0 \le x < 5$$
$$ 0.4 - 0.04x \space 5 \le x < 10 $$
$$ 0 ...
0
votes
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 ...
1
vote
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 < ...
0
votes
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
$$
\gamma(s,x) ...
1
vote
0answers
81 views
Bayesian posterior with integrals over normal densities
Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
2
votes
0answers
33 views
Average Bhattacharyya distance
Two conditional PDF's $f_1(y|X=x)\sim N(Ax,\sigma^2)$ and $f_2(y|X=x)\sim N(-Ax,\sigma^2)$ are given. I need to find their average Bhattacharyya parameter assuming $X$ has a Rayleigh distribution ...
1
vote
1answer
59 views
Calculate expected values for a normal distribution
Suppose that X ~ N(1,2)
Find:
$E(X-1)^4
and E(X^4)$
I have no idea how to get started. Do I just need to integrate the pdf of the normal distribution multiplied by what is between the brackets? If ...
0
votes
1answer
29 views
Find the moment generating function
$$f(y)=\frac{e^{-|y|}}{2}$$
I tried calculating it by integrating $(1/2)e^{ty}\cdot e^{-|y|}) dy$
and splitting up that integral into two separate integrals. However, I did not get a finite answer. ...
2
votes
1answer
41 views
Evaluate $Pr[Y - X < c, X \ge 0]$
Let $f(x,y) = Pr[X=x, Y=y]$ be the joint density of two random variables, $X$ and $Y$.
I have been given s = $Pr[Y - X < c, X \ge 0] = \int_0^\infty \int_{-\infty}^{c+y} f(x,y) dx dy $
I am a ...
0
votes
1answer
120 views
Probability: Determining Which Phone Plan Is Better
A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of $10$ cents per minute, whereas the second charges a flat rate of $99$ cents for calls up ...
1
vote
0answers
47 views
Expected value of threshold function
Let $X$ be a random variable with probability density function $f(x)$, $\alpha>0$ a constant and $g(x)$ a function with $g(x)=\begin{cases}
\begin{array}{c}
g_{0},\\
g_{1}
\end{array} & ...
1
vote
1answer
40 views
Finding $EX$ of a density function (integrating $\ln u$ over infinity)
I've been given a density function as:
$f(x) = 1/4e^{-|x|/2}$ where $-\infty < x < \infty$
and need to show that $EX = 0$
I understand that to find $EX$ I must calculate $\int xf(x)~dx$ ...
0
votes
1answer
89 views
Erroneous Answer Key?
The problem I am working on is:
The current in a certain circuit as measured by an ammeter is a continuous random variable $X$ with the following density function: $f(x) = .075x + .2$ for $3 \le x ...
1
vote
1answer
167 views
Taking the derivative of definite integral?
I'm having trouble understanding the derivative of definite integral. For example, why is the following true?
$\frac{d}{dx}\displaystyle\int_{0}^{x}F_{1}(x-v)f_{1}(v)\, \mathrm{d}v = ...
0
votes
1answer
82 views
How can I solve this integral?
How can I solve the following integral?
$$\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} ...
0
votes
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 ...
2
votes
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 ...
4
votes
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 ...
2
votes
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 ...
0
votes
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 ...
5
votes
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 ...
2
votes
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 ...
0
votes
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 $y$ ...
1
vote
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 ...
3
votes
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 ...
0
votes
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{ ...
3
votes
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:
$$\int_{-\infty}^{\infty} x (F(x + c) ...
1
vote
0answers
71 views
second order stochastic dominance
Let the nonnegative random variables $X$ and $Y$ have distribution functions $F$ and $G$ and density functions $f$ and $g$, respectively.
Suppose $X$ is second-order stochastically dominant over $Y$, ...
0
votes
1answer
65 views
Upper bounds for an integral with an infinite upper limit
I'm trying to work out an upper bound for the following problem, but I'm making very little progress. Hopefully, someone will be able to make a suggestion.
The integral I'm attempting to bound is:
...
2
votes
1answer
86 views
Expectation of a multivariate Gaussian over a plane
For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...
5
votes
1answer
67 views
Evaluating $\int_{0}^{1}\int_{0}^{1}\frac{r^{i+j}(1-r)^{k+l}s^{2m-i-j}(1-s)^{2m-k-l}}{(r+s)^{m}(2-r-s)^{m}}drds$
I'm trying to compute a closed form expression for the integral
$$ \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 ...
1
vote
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 = ...
3
votes
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 $\sigma$.
...
0
votes
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 $x/c$ ...
8
votes
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:
$$
\int_\mathbb{R} x \left( \frac{1}{\sigma_1} \phi\left(\frac{x}{\sigma_1}\right) ...
0
votes
0answers
107 views
Derivative of Riemann integral [of a distribution function].
I am having a trouble solving a derivative of a Riemann integral while trying to obtain a distribution of a variable being a function of another random variable.
Let $ X $ be a random variable with ...
30
votes
4answers
815 views
Rain droplets falling on a table
Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
0
votes
1answer
54 views
Integral of unimodal functions $f>0$.
Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is positive almost everywhere and integrable. We know that if $n=1$ and if $f$ is unimodal then the integral $F(x)=\int_{[-\infty,x]} f$ is convex for ...
4
votes
1answer
102 views
Expectations containing normal CDF
Suppose that $X\sim\mathcal{N}\left(0,1\right)$ (i.e., $X$ is a standard normal random variable) and $a,b,$ and $c$ are some real constant. Does any of the following expectations have a closed-form?
...
7
votes
1answer
105 views
How demonstrate the Craig representation for the Gaussian probability function?
The Q-function is defined by :
$$Q(x) =\frac{1}{\sqrt{2\pi}} \int_{x}^{\infty}\exp(-\frac{u^2}{2}) \ \mathrm{d}u \ \ (1).$$
According to the wiki page there is an alternative form of the Q-function ...
2
votes
2answers
368 views
How to compute this integral involving a cdf?
$\int_0^\infty\Phi(\frac{-x}{\sqrt{2}})d\Phi(x)=?$ where $\Phi(x)$ is the cumulative distribution function of a standard normal random variable.
1
vote
1answer
131 views
Probability that a Multivariate Normal RV lies within a Spherical Region of Radius R
I am currently using different procedures to estimate the probability that a $D$-dimensional Gaussian random variable with mean $\mu$ and covariance $\Sigma$ lies within a sphere of radius $R$ that ...
3
votes
1answer
112 views
The probability of $Ax^2+Bxy+Cy^2 = 1$ defining an ellipse.
In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian ...
4
votes
0answers
126 views
a integral of bivariate Gaussian random variables.
I met the following problem when doing estimation and detection homework.
The problem asks for a maximum likelihood estimator for (v,$\rho$) of bivariate joint Gaussian, where v is the common ...
0
votes
2answers
231 views
Integral of probability density function
$$\int_{-1}^{1}{\int_{x^2}^{1}{cx^2y \ dy \ dx}}$$ Given that:
$\int_{-1}^{1}{\int_{x^2}^{1}{cx^2y \ dy \ dx}}=1$, I have to find $c$. I am a bit disoriented by that $x^2$ as the limit of the first ...
3
votes
2answers
142 views
How can I prove $\int[F(x+a)-F(x)]\,dx=a$
How can I prove $$\int[F(x+a)-F(x)]\,dx=a$$
where $F(x)$ is the cumulative distribution function?
9
votes
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 ...
0
votes
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 ...
