2
votes
3answers
43 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
0
votes
0answers
23 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
1
vote
2answers
28 views

Identifying the distribution which represents a negative binomial distribution as a compound poisson distribution

Suppose that the random variable $X$, which has a negative binomial distribution with probability $p$ and parameter $r$, can be represented as the summation of $N$ iid random variables $Y_1, Y_2, ...
1
vote
0answers
29 views

Probability distribution of k consecutive successes with n maximum trials

Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are ...
0
votes
0answers
37 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
0
votes
1answer
18 views

Show $P(X=n)=\left(\frac{1}{2}\right)^{n+1}$ for Poisson variable with exponentially distributed $\lambda$

I'm supposed to do the following, any help/pointer is appreciated: Suppose $X$ is Poisson distributed with mean $\lambda$. Suppose $\lambda$ is exponentially distributed with mean $1$. Show that ...
0
votes
1answer
34 views

How to increase winning chance in lottery [on hold]

Let us imagine such kind of lottery game :lottery machine is running and randomly is selecting $7$ number from $1$ to $36$(including).out of this $7$ numbers,$6$ are basic or in other word ,jackpot ...
0
votes
2answers
43 views

Existence of density function for a sum of 2 Random Variables

Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist. Does it follow that $Y+X$ has a density function? I am reading ...
1
vote
2answers
36 views

Probability distribution and density of Y=g(X)

$Y=g(X)$ as shown below. Find $f_Y(y)$ and $F_Y(y)$ in function of $f_X(x)$. I began with writing $Y=g(X)$ as the following piecewise function: $ Y = \begin{cases} \ -b & \text{if } x ...
2
votes
2answers
72 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
0
votes
0answers
18 views

Probability Density Function for Discrete and Continuous Random Variables.

From the lecture on Khan's Academy. While discussing the Discrete Random Variable, consider the following RV P[X=0] = 1/4; P[X=1] = 1/2; P[X=2] = 1/4; And after ...
1
vote
0answers
22 views

Conditional return time of simple random walk

Consider a simple random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0 = 0$. The probability to jump to the right neighbour is $p \geq \frac{1}{2}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, ...
0
votes
0answers
25 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
1
vote
1answer
33 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...
0
votes
1answer
26 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
2
votes
1answer
64 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
0
votes
1answer
36 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
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 ...
3
votes
1answer
59 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
0
votes
0answers
27 views

Distributions with infinity variance.

I'm looking for a list (or something like that) of distributions with infinity variance (or infinity second moment), like non-gaussian Stable Distributions. I have an important warning: Some ...
3
votes
1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
0
votes
0answers
45 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
1
vote
0answers
15 views

Are functions of independent random variables related to each other by a constant independent

I have $6$ random variables $a,b,c,d,f,g$, each having independent Gaussian distribution. Now I define following three random variables \begin{equation} X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ...
1
vote
3answers
74 views

Finding expected value??

In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the ...
1
vote
0answers
30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
1
vote
1answer
26 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
0
votes
2answers
23 views

Expected Value with Parameter p

The random variable X has the following probability distribution: P[X=-1]= (1-p)/2 P[X=0]= 1/2 P[X=1]= p/2 The parameter p satisfies the inequality $0 < p < 1$. Find the expected value and ...
0
votes
1answer
37 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
0
votes
1answer
39 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
0
votes
1answer
18 views

Covariance of random variables with identical distribution.

Let $X_1,...,X_n$ be random variables with identical distribution, and for all $i=1,...,n$ $\mathrm{Var}(X_i)$ exist. 1. Show that the covariance between each two random variables exist. 2. Show that ...
1
vote
1answer
33 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...
1
vote
0answers
51 views

Conditioning on function of random variable and random variable itself

Suppose that $Y_{i}\in\{0,1\}$ is a binary variable, and $X_{i}$ is some random vector in $\mathbb{R}^{d}$ . Why can we say the following: \begin{eqnarray*} ...
0
votes
1answer
33 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
2
votes
1answer
36 views

Why is this distribution Poissonian?

Do this experiment. Draw 10000 random number in $[0,1]$ according to the uniform distribution. Order them in the increasing order. The difference between two neighbouring numbers follows a Poisson ...
0
votes
0answers
39 views

How to show $ P\big(\big|\frac{X}{n}-p\big|>a\big)\le\frac{\sqrt{p(1-p)}}{a^2n}min\big\{\sqrt{p(1-p)},a\sqrt{n}\big\}$

Let $X$ be binomial, $B(p,n)$ with $p>0$ fixed, and $a>0$. Show that, $\displaystyle ...
-3
votes
1answer
58 views

Find the expectation $E[X]$ [closed]

Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less then 100. Find the expectation $E[X]$?
0
votes
0answers
43 views

What is the probability density function of the cosine of a gaussian random variable?

I want to find the probability density function of $Y=\cos(X)$, where $X\sim N(\mu, \sigma^2)$. The answer is known when $X$ is uniformly distributed $U(-\pi, \pi)$ and it is an arcsin pdf, given by, ...
0
votes
1answer
41 views

subscript notation in conditional probability

$X$ and $Y$ are two discrete random variables with joint p.m.f $p_{XY}$ such that $p_{XY}(x_i,y_j) = P(X=x_i, Y=y_i)$. I came across a notation that refers to $p_{X}(x|y)$. How do I express it in the ...
-1
votes
1answer
30 views

How to see (in)dependence of random variables based on their joint density

This is a valid joint pdf. I just want to know if X1 and X2 are dependent or independent rvs ? Why ? Thank you for your help. Is there a way of seeing this without computing the marginal density ...
0
votes
1answer
28 views

number of ones with neighbours in a random binary string

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts ...
0
votes
0answers
18 views

A variation of Polya's urn

Polya's urn model is as follows: you have $a\in \mathbb{Z}_{>0}$ red balls and $b\in \mathbb{Z}_{>0}$ blue balls in a urn. Suppose you pick a red ball. Then you put back $c\in ...
1
vote
1answer
92 views

Can someone please help to understand the following probability

I was reading something on communication, then I came across the following equation: $Power_{rx}=Power_{tx}*|R|^2/(1+d^2)$ where $Power_{tx}$ and $d$ can be assume to be constant, and R is the ...
1
vote
2answers
36 views

Probability Generating Functions with Three Dice

Three identical dice are thrown. The dice are fair, that is, for all three dice the probability of turning up face $j$ is $1/6$, $1 \le j \le 6$. Let $X_1,\ X_2,\ X_3$ be the independent random ...
0
votes
0answers
32 views

Expectation and Variation of dependent RVs

This is a really nice question, and while I can think of a solution to both parts, I wonder if there's a more elegant one to the latter: A fair 6-faced dice is tossed once. In a box there are 6 ...
1
vote
1answer
42 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
0
votes
1answer
63 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
1
vote
0answers
38 views

Moment-generating function of a generalised normal random variable

Let $X$ be a random variable that follows the "version 1" generalised normal distribution described here, with p.d.f. ...
3
votes
1answer
169 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
2
votes
0answers
27 views

Rosanov - Probability Theory Chapter 4 Question 5

I am trying to solve one of the questions in Rosanov - Probability (Chapter 4 Question 5), but I am not exactly sure what the question is asking of me. The question is: Random variable $E$ with ...
0
votes
0answers
17 views

The distribution of minmax and maxmin deviations of a Random variable

Let $X_1,X_2,X_3,......,X_n$ be $n$ independently and uniformly distributed random variables in the interval $[a,b]$. Further let $P=\min \{X_i,i=1,2,3..,n\}$ and $Q=\max\{X_i,i=1,2,3..,n\}$. ...