1
vote
0answers
17 views

Easy question: Multiple random variables vs. product of probability spaces

I never had a course in probability theory and the definitions we work with are quite informal, so I am a little bit confused about the difference between "multiple" random variables and the notion of ...
0
votes
1answer
10 views

Limiting random variable

My question is about limiting r.v. Suppose, we have a sequence of r.v.s. $\{X_n\}$. And we know that $\liminf X_n=-\infty$ and $\limsup X_n =\infty$ almost surely. What can we say about $\lim X_n$. ...
0
votes
0answers
14 views

Density function of $Y $given that $ Y = 2X, X \leq 2, Y = X^2, X > 2.$

So we have $X$ with density $$f_X(x) = 1/x^2, x \geq 1, f_X(x) = 0, x < 1.$$ And $$Y = 2X, X \leq 2, Y = X^2, x > 2.$$ So I drew my graphs of $f_X$ and $Y$, but where the functions change is ...
-1
votes
1answer
25 views

Question in permutations

When we use this law? And in any case we use it? Thank you and I wish clarification.
0
votes
1answer
33 views

Battery lifetime as normal distribution?

I want to model battery lifetime, which decrements continuously at every epoch (i.e., work-cycle) in the following way. So it takes values such as 100, 99.7, 99.3, 99.2, ... 0 (a continuous random ...
5
votes
2answers
67 views

“Back to square one” problem

There's a problem I've been stuck on in preparation for junior programming contest I'm going to participate in. It is as follows: The "back to square one" problem is played on a board that has ...
0
votes
1answer
17 views

what is the physical significance of probability density fnction

In literature I've studied that the probability density function of a random variable X is the derivative of probability distribution function. Mathematically speaking, it is the slope of the ...
1
vote
1answer
24 views

Distribution of things on one row.

What is the meaning 4|5 4|3 2|1 I think it (4 boys x (2 girls x 3 boys) x (4 boys x(2 girl ... I wish that I understand so well and thank you.
1
vote
1answer
39 views

If $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$

If $X\geq 0$, and $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$. We know that that $$\mathbb{E}(\min(X,t))=\int_{X\leq ...
0
votes
0answers
20 views

Finding a copula of beta distributions

Consider two given beta-distributed random variables $X_1, X_2$ with cumulative distribution functions $F_1, F_2$ and a given value of "dependence" $\rho \in [-1,1]$. I want to find a copula $C$ for ...
2
votes
1answer
23 views

Analogous of Markov's inequality for the lower bound

Consider a positive random variable $X$ and call $E[X]$ its expectation. For any positive $a \in \mathbb{R}$, an upper bound for the probability of $P(X>a)$ is provided by the Markov's Inequality, ...
1
vote
0answers
23 views

Probability question on 5 components operation after a given time

My attempt: $\lambda=1/2.5=0.4$ Since $P(T\geq t)=({1-e^{-\lambda t}})$ $P(T\geq3)=({1-e^{-0.4(3)}})^5$ However my book says: $P(T\geq3)=({e^{-0.4(3)}})^5$ Why is this? How did the book do ...
1
vote
1answer
36 views

Find conditional probability $\mathbb{P}(X \le x | \max(X,Y)) $

Let $X,Y$ be iid such that $X\sim F>0$ and $Y \sim F>0$ ($X$ and $Y$ have the same probability distribution). Find $\mathbb{P}(X \le x | \max(X,Y)) $. I know that $\max(X,Y) \sim F^2$. I ...
-1
votes
0answers
23 views

Stationary distribution of a birth-death model where a parameter follows a uniform distribution.

I asked this question about some type a markov process I was interested in. @Did offers an answer but I fail to understand how to apply his answer to a concrete example. I am therefore seeking for an ...
0
votes
1answer
32 views

How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
0
votes
1answer
23 views

how many types of events actually exists in the theory of probability?

I read many article on the internet and found that there are only three types of event that can be occurred(or that has been considered in the probability theory). those are : mutually exclusive ...
0
votes
1answer
16 views

Jump Set v. Range of Randome Variable

What is the difference between the range of a random variable X, and its jump set? I know that they are not equivalent sets, e.g. for a continuous RV, the range is $(- \infty , \infty)$, but the jump ...
1
vote
2answers
25 views

Distribution of a distance between random numbers

I'm working on a problem in which I came to a question concerning distribution law of a result of operations on random variables. I will ask a simple question and hope to understand the concept from ...
0
votes
1answer
18 views

What is the sum-capacity for a non-symmetric interference channel for information theorists?

This question is dedicated for people who are experts in information theory. An interesting result for a two user interference channel in information theory, is the sum-capacity to within one bit. It ...
-3
votes
0answers
44 views

$ \ P(X_1<X_2<X_3|X_3<1)$ given $f(x) = e^{-x}$ [closed]

Find the probability$ \ P(X_1<X_2<X_3|X_3<1)$ Given: $f(x) = e^{-x}, \ 0<x<\infty$ zero elsewhere. The variables $ x_1,x_2x_3$ have same pdf and they are independent variables. Here is ...
0
votes
0answers
24 views

Probability of an event happen using a uniform number?

I am starting to work in a mathematical modeling in agriculture field, and I am not a statistician or mathematician. Could you please help me to explain this? I have a curve of "probability of ...
0
votes
0answers
40 views

Conditional mean: E(Y|x)

Please help.I am not sure with my answer.Anyways, the problem goes this way: Find the conditional mean of $Y$ given $X=x$ ,$E(Y|x)$, if X and Y have the joint pdf of $f(x,y)=21x^2y^3, ...
-1
votes
1answer
32 views

Probability of $P(X_1<X_2|X_1<2X_2)$

Find the $P(X_1<X_2|X_1<2X_2)$ Given: $$f(x) = e^{-x}, \qquad 0<x<\infty$$ zero elsewhere. The rvs have same pdf and they are independent variables. Here is my attempt: ...
0
votes
1answer
16 views

Distribution function and probability of random variable R with density function $f(x) = 1/2e^{-|x|}$

So we have a random continuously variable $R$ with density function $f_R(x) = 1/2e^{-|x|}$. First I need to sketch the distribution function of $R$. So $$F_R(x) = \int_{-\infty}^x f_R(x) dx$$, but ...
0
votes
3answers
53 views

question at Probabilities [closed]

In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the ...
-1
votes
2answers
30 views

How is the binomial distribution connected with the theoretical approach to probability?

I've been told the theoretical approach to probability is defined as follows $$\operatorname{Pr}(\textsf{something})=\frac{\textsf{Favorable events}}{\textsf{possible events}}$$ This has to be ...
1
vote
1answer
30 views

Proving a simple aspect of the CDF for discrete functions F(b) - F(a) [closed]

Let $F_X$ be the cumulative distribution function for discrete random variable $X$. How would you prove this: $$ F_X(b) − F_X(a) = \operatorname{Pr}(a \lt X \le b) $$
0
votes
0answers
23 views

Convolution of two random vectors

Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where $\mathbf{y}=A \ast B$, here $\ast$ ...
0
votes
1answer
30 views

conditional distribution of X, given Y=y and Z=z, and compute E(X|y,z)

Given: $$f(x,y,z) = \frac23 (x+y+z), \,\,\, 0<x<1,\,\,\, 0<y<1,\,\,0<z<1$$ zero elsewhere. Find the conditional distribution of $X$, given $Y=y$ and $Z=z$, and compute $E(X|y,z)= ...
1
vote
1answer
37 views

Probability of $P(X_1 X_2\le 2)$

What is the probability of $P(X_1 X_2\le 2)$. Both variables are independent and each has the probability density function $f(x)=1, 1<x<2$, zero elsewhere. First I would like to assume that the ...
0
votes
0answers
43 views

How to prove that $X+Y \mod p$ is indpendent from $X$ if $X$ and $Y$ are independent?

We have a group $\mathbb{Z}_p$ and some random variable $X$ and $Y$ with this domain. We have that $Y$ is chosen uniformly at random, thus each element from $\mathbb{Z}_p$ with probability ...
0
votes
1answer
24 views

Finding a CDF from the PDF of another Random Variable

Given a function $$ Y=ae^x $$ With the distribution: $$ f_X(x)=be^{-bx} \,\,\,\, x\geq0 $$ Show that the cumulative distribution function is: $$ F_Y(y)=1-(y/a)^{-b} \,\,\, y\geq c $$ My approach ...
-1
votes
1answer
43 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
0
votes
0answers
29 views

Creating a random network (graph) with a $\textbf{random}$ number of vertices and given degree distribution

I was trying to find an answer to my question on google scholar, however I didn't find anything that is close to what I am looking for. I would be very grateful for your help. There is a theory of ...
0
votes
1answer
15 views

Bounds of integral in Power function

Here is the question: Let $X_1,X_2$ be iid uniform $(\theta,\theta+1)$. For testing $H_0:\theta=0$ versus $H_1: \theta>0$, we have two competing tests: $\hspace{15mm}\phi_1(X_1):$Reject $H_0$ if ...
1
vote
1answer
34 views

MLE of $n(\theta,a\theta)$ family

Question: A special case of a normal family is one in which the mean and variance are related, the $n(\theta,a\theta)$ family. If we are interested in testing this relationship, regardless of the ...
1
vote
2answers
43 views

Should I avoid distribution functions in probability?

I'm reading Erhan Çınlar's book on Probability and Stochastics, and in Chapter 2, he says that distributions are used extensively in elementary probability theory in order to avoid measures. And ...
0
votes
1answer
24 views

What is application of gamma distribution on pure math or probability theory?

What is application of gamma distribution on pure math or probability theory? i saw it on several probability textbook as a definition, but it seems to me mathematician couldn't derived it if it is ...
0
votes
0answers
17 views

What is the joint probability density function

I want to find the joint PDF of vector $\mathbf{z}$ where $z_i=|y_i|^2, i=1 \dots N$ and $y_i$ is $i^{th}$ element of $\mathbf{y}=\begin{pmatrix} a_{1,1} & 0 & 0 & 0 &\cdots & ...
0
votes
1answer
17 views

Is truncating a discrete probability mass function possible?

I have random variable X, and probability distribution: $P[X = A] = .4$ $P[X = B] = .3$ $P[X = C] = .2$ $P[X = D] = .1$ I want to create a conditional probability with event F. Where F is the ...
-2
votes
1answer
66 views

Is $\theta_1-\theta_2$ independent of $\theta_1-\theta_3$ given all are uniform random variables between $[-\pi,\pi]$

I have three random variables $\theta_1, \theta_2, \theta_3$ all are i.i.d uniformly over $[-\pi,\pi]$. These in reality represent angles in my problem that I am trying to solve. I have a linear ...
5
votes
1answer
96 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
1
vote
1answer
35 views

The smallest integer $n$ for a Poisson distribution

Along a stretch of motorway, breakdowns require the summoning of the breakdown services occur with a frequency of 2.4 per day, on average. Assume the breakdowns occur randomly and that they follow ...
1
vote
0answers
136 views

Does clever noise exist?

This question is about a random noise, which is called "clever" if its distribution function satisfies a certain condition. Let $Y_0$ and $Y_1$ be two continuous random variables on the same ...
0
votes
2answers
18 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
1
vote
1answer
11 views

Find distribution of a bernoulli funtion of a unifrom random variable?

I have a uniform random variable $\theta \in [-\pi,+\pi]$. I also have a bernoulli function of this random variable $G(\theta)$, defined as follows, \begin{align} \begin{cases} 1 & \text{if $ - ...
0
votes
1answer
30 views

Dirichlet distribution, sum of Beta distributions

I currently have a problem about Dirichlet distributed Variables. In one of the papers I am currently reading it says: Let $S=(S_1,...,S_m)\sim Dir(\delta\omega_1,..., \delta \omega_m)$, with ...
0
votes
0answers
27 views

conditional expectation of two independent normal random variable

Show that conditional expectation of two independent normal random variable $(X,Y) \approx N(m=(m_1,m_2), \Sigma)$ is equal: $$E[X|Y] = m_1 + \rho \frac{\sigma_X}{\sigma_Y}(Y-m_2)$$ Is there any way ...
0
votes
0answers
15 views

Expectation of truncated negative binomial distribution

This is what I have done, I would like to know if there is a way to simplify it, solve summations,...
0
votes
3answers
33 views

Calculate the PMF, mean and variance of X for x=-1,1

An Urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X=1 if a red ball is drawn, and let X=-1 if a white ball is drawn. Give the pmf, mean, and Variance of X. I know ...