1
vote
1answer
16 views

Probability of a single variable from a Moment Generating Function

This is from the A/S/M study guide, and the answer is listed, I just don't understand how he's arriving at the answer... I'm sure it's something simple I am missing! I have two identically ...
0
votes
1answer
63 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
4
votes
2answers
93 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
1
vote
1answer
62 views

Question about transformations and sums on uniformly distributed random variables.

I'm looking into a few problems as a hobby of mine, and found myself with the following problem: let $X$ be a random variable uniformly distributed on $[0,1]$. What is the probability that after $N$ ...
4
votes
1answer
62 views

What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
0
votes
2answers
66 views

Pdf of $Z=(XY)^{1/2}$. with X,Y independent r.v. with the same distribution (iid) [closed]

Let be $X,Y$ two independent random variables having the same distribution (the following is the density of this distribution) $$f(t)= \frac{1}{t^2} \,\,\, \text{for $t>1$}$$ Calculate the ...
2
votes
2answers
48 views

What's the density of $Z=\max(X,Y)-\min(X,Y)$ with $X,Y$ exponentials of parameter $\lambda$?

Let be $X,Y$ two independent exponential random variables with parameter $\lambda$. What is the pdf of $Z=\max(X,Y)-\min(X,Y)$? Thanks for your help.
1
vote
3answers
62 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
0
votes
0answers
30 views

Convolution of exponential family distributions

Is there a general form for the convolution of two exponential family distributions?
2
votes
0answers
48 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
0
votes
1answer
39 views

Difference between two independent geometric random variables

Let $\xi_1$ and $\xi_2$ be independent random variables: $\xi_1 \simeq Geom(1/2), \xi_2 \simeq Geom(1/6)$. How do you find the probability mass function of $\eta=\xi_1-\xi_2$ using convolution?
0
votes
0answers
104 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
0
votes
1answer
19 views

Fidelity of measurement using conditional probabilities

Let me begin by saying that I'm not entirely sure if this is the correct forum, or if Cross Validated would be more suitable. The problem I'm about to describe is statistical in nature, but I believe ...
1
vote
1answer
77 views

Sums of normal CDF's

This is my following problem: $$ CDF_A:F_A(x)=\Phi(x)^2 $$ $$ CDF_B:F_B(x)=1-\Phi(-x)^3 $$ $$ Defining: X=A+(-B) $$ I have those two CDF's and I want to calculate the probability that X is smaller ...
1
vote
1answer
32 views

Sums of random variables expressed as convolution

If $X_1, \dots, X_n$ are a sequence of IID random variables, and $S_1, \dots, S_n$ is the sum of the first n random variables, i.e. $S_1 = X_1$ and $S_n = \sum_{i = 1}^{n} X_i$. Apparently, we can go ...
0
votes
2answers
33 views

Third Moment of a Sum of Normal and Gamma

I just ran into the next problem: The random variables $X$ and $Y$ are independent, where $X \sim Normal(1,1)$ and $Y \sim Gamma(\lambda,p)$ with $E(Y) = 1$ and $Var(Y) = 1/2$ How do we find ...
1
vote
0answers
91 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, \, ...
0
votes
0answers
54 views

Probability that two bivariate distributions differ

My problem is similar to the unanswered Probability of collision (two bivariate normal distributions): I have two points in 2D space, let's say $A(\mu_x, \mu_y)$ and $B(r_x, r_y)$. The uncertainties ...
1
vote
0answers
29 views

$f_{X^2}(x)$ VS $f_X(x^2)$ [duplicate]

Sorry, this time the format should be accurate. In probability, when we try to describe a pdf, we write it as $f_X(x)=1/x$, which means the random variable is X and the x is the specific variable in ...
1
vote
1answer
146 views

Convolution of indicator functions is continuous

Suppose I have an indicator function on a set of measure $E$, which is a subset of $[0,1]$. Is the function of this indicator convoluted with itself a continuous function? How can I show that it is? ...
0
votes
1answer
148 views

Probability with bullets and walls

There are two shooters with different guns and bullets. Each shooter shoots a bullet to a different target hanging on a wall. The hit of each bullet follows a normal distribution centered on its ...
0
votes
1answer
46 views

Maximum of one exponential and one uniformly distributed random variable

If X and Y are independent random variables with X exponentially distributed with mean 1 and Y uniformly distributed in [0,1] , how do I find the distribution of Max(X,Y)
1
vote
1answer
55 views

What is the distribution of $X+Y$ where $X \sim U(0,\frac{L}{2})$ and $Y \sim U(\frac{L}{2},L)$?

I started along these lines: Let $Z = X + Y$ where $\frac{L}{2}< z < \frac{3L}{2}$, then, $$f_{X+Y}(z)=f_{Z}(z) = \int f_{X}(x)f_{Y}(z-x)dx$$ However, I am not sure how to fill in the bounds ...
-2
votes
1answer
173 views

Convolution of Discrete Uniform ,$DU$, Distribution.

If $X\sim DU(k,a,h),\quad -\infty<a<\infty,h>0=1,2,\ldots$ then the probability function is $$P(X=a+jh)=\frac{1}{k},\quad j=0,1,\ldots,k-1$$ Let $Z\sim DU(r,0,s)$ and $Y\sim DU(s,0,1)$ , ...
0
votes
1answer
129 views

Convolution of two dimensional gaussian functions

I want to calculate the sum of two probability density functions. I know that it is: $P_{U+V} (x)= (P_{U} * P_{V})(x)$ If $P_{U}$ and $P_{V}$ are gaussian functions in one dimension, i.e. $P_{U}(x) ...
0
votes
1answer
98 views

Analytic solution of the convolution of two discoutinous c.d.f s

I have a c.d.f of variable X with a mass point at the end point, $$F(x) = \begin{cases} 0 & x<a,\\ 1-\frac{m}{x+m-a} & a\le x < r-a,\\ 1 & x\ge r-a. \end{cases} $$ where m>0. Is it ...
1
vote
1answer
377 views

Finding distribution of $X^2+Y^2$ where $X,Y\sim N(0,1)$

Assume I have two random independent standard normal variables $X,Y\sim N(0,1)$, How can I find the distribution of $Z=X^2+Y^2$? I thought integrating the convolution, i.e ...
2
votes
1answer
77 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
3
votes
2answers
1k views

convolution of exponential distribution and uniform distribution

Given $X$ an exponentially distributed random variable with parameter $\lambda$ and $Y$ a uniformly distributed random variable between $-C$ and $C$. $X$ and $Y$ are independent. I'm supposed to ...
2
votes
1answer
161 views

What is the distribution of empirical covariance between two independent normal distributions?

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
1
vote
1answer
328 views

convolution square root of uniform distribution

I need to find a probability distribution function $f(x)$ such that the convolution $f * f$ is the uniform distribution (between $x=0$ and $x=1$). I would like to generate pairs of numbers with ...
0
votes
1answer
203 views

what's the distribution of the inverse of a random variable that follows a negative binomial distribution?

I was studying the method of moments estimation of parameters, and I encountered the following problem. I have a geometric distribution as following: $P(X=k) = p(1-p)^{k-1}$, and a sample size of n, ...
0
votes
1answer
129 views

Probability density of vector sum

Consider two unit $\mathbb R^2$ vectors $v$ and $w$. Then $v+w$ lies within a (closed) circle with radius 2, that is, in the region $x^2+y^2\leq4$. Intuitively, the probability of $v+w$ lying close ...
0
votes
0answers
255 views

Convolution of two functions (pdfs)

I want to convolve two signals . The range of each of the signal is 0 to 1. ...
1
vote
0answers
63 views

Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
1
vote
1answer
326 views

Computing convolution density function with Maple

I would like to know how to compute a probability function of a convolution of Negative Binomial distribution with Maple. Here is an easy example of what I want to do : ' ...
2
votes
1answer
1k views

The distribution of the sum of two independent exponential distributions

I am trying to calculate the distribution of the sum of two independent log-uniform distributions but something doesn't add up. Suppose $a \sim \mathrm{uni}(0,1)$ and $b \sim \mathrm{uni}(0,1)$. ...
0
votes
3answers
278 views

Derivation of pmf from convolution

Suppose that a discrete random variable (with finite support) $Y$ is given by $Y = X_1 - X_2$, where $X_1$ and $X_2$ are both discrete random variables with finite support and with the same ...