Tagged Questions

Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

learn more… | top users | synonyms

0
votes
0answers
2 views

ranndom process and probabiity

Let X and Y be two independent zero mean Gaussian random variables with variances 𝜎𝑋 2 = 1, 𝜎𝑌 2 = 2 . Find the pdf of the random variable Z=|X+2Y| and from it compute the mean and variance of Z ...
0
votes
0answers
12 views

Poisson Distribution word problems

During rush hour the number of cars passing through a particular intersection23 has a Poisson distribution with an average of 540 per hour. (a) Find the probability there are 11 cars in a 30 second ...
0
votes
0answers
10 views

Find the distribution function for random vector and what if it has discontinuities

Let $X,Y$ be random variable. Suppose $Z:=(X,Y)$ be random vector and $P(X=Y)=1$ and $P(a \le X \le Y) = b-a$ with $0 \le a \le b \le 1$ Question 1: what is the distribution function of $Z$? ...
1
vote
3answers
34 views

Find the pdf of T = X + Y

Let (X,Y) be a random point chosen uniformly on region R = {(x,y) : |x| + |y| <= 1}. I need to find the pdf of T = X + Y. I know the joint density is just equal to 1/(area) = fxy(x,y) = 1/2 for ...
1
vote
0answers
30 views

Probability Distribution for a Weird Card Game

I promise this is not for a homework problem, even though this sounds like only something a professor would dream up. Here is the game: I begin with a deck of 13 cards: 1 through 10, Jack, Queen, and ...
1
vote
0answers
19 views

Gaussian random vector with 0 mean [duplicate]

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
2
votes
1answer
10 views

How is this Variance found in this old question?

On this question: Probability: Normal Distribution they find these values: $\hat\mu = .05(150) = 7.5\space,\hat\sigma = \sqrt{150(.05)(.95)} = 2.67$ I see how they got $\mu$, but how did they get ...
-1
votes
0answers
16 views

Lottery Ticket Probability [on hold]

At a certain retailer, purchases of lottery tickets in the final 10 minutes of sale before a draw follow a Poisson distribution with mean = 15 if the top prize is less than 10,000,000 and follow a ...
-2
votes
0answers
10 views

Random Sample taken [on hold]

A random sample of 300 people are taken. What is the probability that at least 100 of them are over 180cm in height given average height = 175 and standard deviation = 10?
0
votes
0answers
10 views

Right continuity of right inverse of right continous map

I am stuck on the following proof that I found in Dellacherie-Meyer's book "Probabilities and potential B", p. 119 (increasing processes and projectors). Given a map $a$ on $[0,\infty [$ which is ...
2
votes
1answer
11 views

distribution of $\|P_VX\|^2$ with orthogonal projection $P$ onto $V$

We've had the following question discussed today but without any result: Let $X_1,\dots,X_d$ be random variables, iid and $X_n\sim N(\mu_n,1)$. How can we describe the distribution of $\|P_VX\|^2$ ...
1
vote
2answers
30 views

Can 2 different random variables have the same CDF?

I'm looking for proof that two different random variables can have the same Cumulative Distribution Function; in other words, I'd like to disprove that a CDF uniquely defines a random variable. ...
0
votes
2answers
23 views

Co-relation Coefficient

$X$ and $Y$ are jointly continuous random variables. Their probability density function is: $$f(x,y) = \begin{cases}2x & \mbox{if } x\in [0,1], y\in[0,1] \\ 0 & \mbox{ otherwise ...
2
votes
1answer
22 views

Exponential distribution - maximum earthquake magnitude

Suppose $n$ earthquakes occur, and suppose the magnitude of earthquakes are independent and have an exponential distribution with mean $1$. What is the pdf of the maximum earthquake magnitude?
4
votes
0answers
84 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
1
vote
1answer
26 views

Joint probability density function probability

$X$ and $Y$ are jointly continuous random variables. $$f(x,y)=\begin{cases}kx & x\in[0,1], y\in [0,1]\\0 & :\text{otherwise}\end{cases}$$ a) What value of $k$ makes this a density ...
0
votes
1answer
19 views

Represent probability with multiple distributions. Archer shooting bullseyes problem.

The goal is to come up with two ways to represent this probability: An archer shoots a bulls-eye with probability $0.4$. If the archer shoots ten arrows, what's the probability that at least 3 are ...
2
votes
1answer
32 views

Find the value of k which makes f a density function.

Observe the following probability density function for a continuous random variable X $$f (x) = \begin{cases} k\sqrt x (1-x) &\text{ for }x\in(0,1)\\ 0 &\text{ otherwise} \end{cases} $$ Find ...
2
votes
2answers
20 views

Expected value of X-x for exponential distribution

Assume $X\sim$ exponential$(\lambda)$. In class we noted that $E[X-x|X\geq x]=\frac{1}{\lambda}$. Why is this? I would have thought that $E[X]-E[x]=\frac{1}{\lambda}-x$.
0
votes
1answer
31 views

Drawing Probability Density Function

Can someone help me to draw this pdf? I really don't have an idea how to convert a function to pdf. Thank you p(x | c) = 1/3 for 1 <= x <= 4 and P(c) = 0.5
3
votes
2answers
63 views

Normal Distribution and Cofffee

For my homework I have this question: A coffee vending machine automatically pours different types of coffee into cups. The amount of coffee dispensed is modeled by a normal distribution with mean ...
1
vote
0answers
29 views

Probabilty Models and distribution techniques

Coliform bacteria are distributed randomly and uniformly throughout river water at the average concentration of one per twenty cubic centimeters of water. Part (c) In testing for the concentration ...
1
vote
2answers
25 views

For Continuous RVs $X$ and $Y$ if $Y=$ “the CDF of $X$ evaluated in $X$” why does that mean $Y$ is uniform over (0,1)?

Let $X$ be a continuous random variable with CDF $F$. Define the random variable $Y=F(X)$, show that $Y$ is uniformly distributed over $(0,1)$. I have literally no idea why this would even be ...
0
votes
0answers
8 views

Help in finding the distribution in embedded space

The model generating the observation is of the form $y_n = A^Tx_n + U2_n$ where $x$ is the output of a a linear stationary model and $U2$ is a zero mean Gaussian noise of known variance. Now, ...
0
votes
0answers
15 views

Questions dealing with Poisson distribution technique

"A waste disposal company averages 6.5 spills of toxic waste per month. Assume spills occur randomly at a uniform rate, and independently of each other, with a negligible chance of 2 or more occurring ...
1
vote
2answers
29 views

Compute the mean and variance given a probability mass function

I'm given the formula: $$(1-p)^{x-1}p, x = 1, 2, \ldots, \infty$$ and I'm asked to find the mean and variance. I know the mean is represented by $\sum_{i=1}^n p_ix_i$ and the variance by ...
0
votes
1answer
18 views

Discrete random variables and probability models

During jury selection a large number of people are asked to be present, then persons are selected one by one in a random order until the required number of jurors has been chosen. Because the ...
1
vote
2answers
39 views

Determining the Probability Distrubitive Function

A coffee chain claims that you have a 1 in 9 chance of winning a prize on their “roll up the edge” promotion, where you roll up the edge of your paper cup to see if you win. If so, what is the ...
0
votes
1answer
18 views

Random Distance on Torus

Let $U=(X_U, Y_U)$ and $V=(X_V, Y_V)$ be two independent random points in $[0,1] \times [0,1]$, where each possible position is equally likely. Now I am interested in the probability that these two ...
0
votes
1answer
24 views

Statıstıc problem

Will I use binomial distribution for this question? Can you help me please thnk you
0
votes
0answers
16 views

Is the first partial derivative of the cumulative joint density function equal to the marginal density

Is the first partial derivative of the cumulative joint density function $F(X,Y)$ equal to the marginal density? That is, is $\partial F(X,Y)/\partial X = f(X)$? What's the second partial derivative ...
0
votes
1answer
29 views

Probability Mass fuction for scratch ticket

A lottery ticket has 4 squares, each with either a star or an X. The printer printing these tickets has a 20% chance of printing a star per individual square. If the amount paid per star is as ...
0
votes
0answers
18 views

Statistic problem

Can you help me to solve this problem pls,I have exam and I am studyıng. What wıll I use, bınomial or Other thing ? Thank you
0
votes
2answers
21 views

Statistic binomial dist

Can you help me to solve this question pls, I consider that I Will use binomial distrıbutıon but I couldnt
0
votes
0answers
24 views

Inequality of gamma distribution

Let $X_{\alpha} \sim \text{Gammma}(\alpha,1),\alpha >0$ with distribution function $G_{\alpha}$ and $X_{\beta} <_{(c)} X_{\alpha}\,\forall ~0<\alpha<\beta<\infty$. Then show ...
0
votes
0answers
14 views

Probability distribution vectors over a set $S$

Given a (discrete) set $S$, is there a standard notation for the set of all distribution vectors over $S$? That is a notation for the set $$\{X\in[0,1]^S\mid\sum_{s\in S}X_s = 1\}$$
1
vote
0answers
13 views

Rate of Convergence in CLT for IID random vectors with dependent entries

Okay, so suppose I've got a sequence of IID random vectors $X_{n}$, where the entries of each $X_{n}=(X_{n,1},\dots, X_{n,i})$ are dependent. The motivating example here is the multinomial ...
3
votes
1answer
23 views

find the moment generating function of a pdf

Let $X$ be a random variable with pdf $$f_x(x)=\frac{1}{2\sigma}e^\dfrac{-|x-\mu|}{\sigma}$$, $-\infty< x<\infty$, $-\infty< \mu<\infty$, and $\sigma>0$. I have to find the mgf of $X$?. ...
0
votes
0answers
8 views

How to model and plot non-stationary mean-valued data

What would be the best way to plot a distribution of the data whose mean is non stationary. For example: if I have a data series, say y = [100, 97.3, 95.2, 93.2, 91.1, 91.2, ... 0] which yields ...
0
votes
0answers
29 views

Covariance of two normally distributed random variables

$$X = \frac1N \sum_1^N d_icos\theta_i $$ $$Y = \frac1N \sum_1^N d_isin\theta_i $$ $$d_i \sim LN(m_i, \sigma^2) $$ (LN mean Log normal distribution, $m_i$ can be measured, actually function of ...
1
vote
1answer
16 views

Distribution of orthogonal projection onto $\{(x,\dots,x\}\subset \mathbb R^d$

Let $D:=\{(x,\dots,x)\mid x\in\mathbb R\}\subset\mathbb R^d$ and $X$ be a random variable and normally distributed $X\sim N(\mu,\sigma^2 E_d) $ with $\sigma\neq 0$, $\mu\in\mathbb R^d$ and $E_d$ the ...
1
vote
1answer
19 views

How to express combined discrete-continuous RVs in one pdf?

Let's say we have a random variable $X$ that behaves in two different ways where $X\sim$Bernoulli(1/3) AND $X\sim U(0,1)$. $X$ follows the Bernoulli distribution 25% of the time and the uniform ...
0
votes
0answers
14 views

Singular values of $A$, where elements of $A$ are dependent centered normals

Fix $n \in \mathbb{N}$ and let $v$ be a multivariate normal random vector $v \sim N(0,\Sigma \in \mathbb{R}^{n^{2} \times n^{2}})$. Note that $v$ has $n^{2}$ entries. Then, let $A$ be the $n \times ...
2
votes
0answers
58 views

A hard integral from probability theory

I am trying to resolve this integral, which comes out of considering a compound distribution of normal variables: $$ \int_{-\infty}^{\infty} \frac{1}{\sigma_{\sigma} \sqrt{2 \pi}} ...
0
votes
0answers
24 views

Stochastic processes

Update I am a bit confused whether $y_t$ is independent over time under the following assumptions: Consider, first a RV $A$, that follows this process: $A_t = \rho A_{t-1} + e_t$, where $e_t$ is ...
1
vote
2answers
43 views

Probability in dice rolls

$A$ rolls a standard $6$ sided die $20$ times while $B$ rolls it $21$ times. Find the probability that the $A$'s outcome is more than $B$'s. Here, outcome means the sum of the numbers appearing on ...
1
vote
0answers
18 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
0
votes
4answers
46 views

Expectation of product of cosine and sine

$\theta\sim U(-\pi,\pi)$. When $\theta$ follows uniform distribution, what is the expected value of the producot of cosine and sine, i.e. $$E[\sin\theta \cos\theta] = \ ?$$
0
votes
1answer
22 views

Normal approximations and Binomial distributions

I am having some difficulty with the following question from my textbook. I have really been trying to understand the use of normal and binomial approximations, but I'm getting really confused. Any ...
0
votes
1answer
29 views

How to prove for an arbitrary distributions?

Assume we have some distribution $P(x, y)$ on $\mathbb{X} \times \mathbb{Y}$, how to show that: $$ \mathbb{E}_{x, y} \Bigl[ \bigl( y - \mathbb{E}[y \mid x] \bigr) \bigl( ...