0
votes
1answer
6 views

iid random variables (vectors)

If $(X_{1},Y_{1}), (X_{2}, Y_{2}),...,(X_{n}, Y_{n})$ denote a sequence of iid random variables from $(X,Y)$, can I say that each $X_{i}$ is independent from each $Y_{i}$? Or is it just for the ...
0
votes
0answers
19 views

Conditional Probability Question - on route availability

Hey Guys I am seemingly stumped with this question I have gotten involving conditional probability and routes Suppose route $A$ to $B$ is available 0.5 of the time An alternative route to B from A ...
2
votes
2answers
47 views

Does a proportion have to be a rational number?

Does a proportion have to be a rational number? For example, Assume we have a square with side $2$ units. We are throwing a circle of radius $1$ unit over the square. Let $X$ be the area of the ...
1
vote
1answer
11 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
0
votes
0answers
10 views

linear system output when input is a Gaussian process?

Rectently, I read a technical book that says:" the linear transform of a Guassian process is also a Guassian process. i.e. for continuous time case: $$ x(t)*h(t)=y(t)$$ the input $x(t)$ is a ...
0
votes
0answers
28 views

Order statistics of random variables

Let $\{I_1, I_2, \dotsc, I_N\}$ be $N$ i.i.d random variables. I know that the smallest orders statistics and the largest one are defined respectively as follow: $$I_{(1)}=\min(\{I_1, I_2, \dotsc, ...
1
vote
0answers
45 views

Almost sure convergence of a sum of independent exponential random variables?

I'm in difficult with this exercise... I hope someone can help me. Let $X_1,X_2,...$ be independent random variables, $X_n\sim \exp(\lambda_n)$, where $$0 < \lambda_n\rightarrow \lambda , \lambda ...
0
votes
0answers
20 views

For a random variable $X$, $M_x(t) = (1/81)(e^t+2)^4$. Find $\mathbb{P}(X \leq 2)$.

For a random variable $X$, $M_x(t) = (1/81)(e^t+2)^4$. Find $\mathbb{P}(X \leq 2)$. I know I need to somehow expand the mgf so then I can just sum the coefficients for $e^{0t}, e^{1t}, e^{2t}, ...
1
vote
1answer
40 views

Find the probability generating function

I have an exercise of this type that I just can not solve "Are $x$ and $y$ be independent random variables, $X$-Poisson($a$), $Y$-Poisson($b$). Find the probability generating function of the random ...
1
vote
1answer
43 views

How do we square a random variable?

How do we square a random variable? For example, Let $Y=X^2$. $$f_X(x)={\frac{1}{\sqrt{2\pi}}} \cdot e^{\tfrac{-x^2}{2}}$$ How do we derive $f_Y(y)$? Thanks in advance.
1
vote
1answer
120 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
0
votes
0answers
12 views

a question which is somhow related to law of large number

suppose that p = [p1, p2, ..., pn]' is a random vector. (' == transpose) and each element of p like pi is a Gaussian random variable with zero mean (E(pi)=0) and variance vi (E(pi^2)=vi). the ...
0
votes
0answers
25 views

Extension to the Coupon Collector Problem

If there's n different coupons. Instead of ordering coupons one-by-one until you collect all n coupons as in the traditional 'Coupon Collector Problem', what if the coupons came in packs of m coupons. ...
1
vote
2answers
42 views

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?
1
vote
4answers
247 views

Inferring covariance cov[X,Z] from cov[X,Y] and cov[Y,Z] of known distributions

Suppose X, Y and Z are real random variables of known distributions. If one knows the covariance $COV(X,Y)$ and $COV(Y,Z)$, is it possible to infer $COV(X,Z)$?
1
vote
0answers
38 views

Minimum of N Chi-square random variables when N is large

I have a problem in numerically evaluating the PDF of $Y=\min(X_1,X_2,\cdots,X_N)$ where $N=\binom{M}{K}$, the binomial coefficient and $X_i$s are iid Chi-square random variables. The CDF of $Y$ is ...
1
vote
2answers
39 views

Product & Ratio's of 2 Random Variables

I'm interested to know whether it's the case that for random variables $X$ and $Y$ whether or not the ratio of $X$ and $Y$ can be computed as the product of $X$ and $1/Y$. That is, Is $\frac{X}{Y} ...
0
votes
1answer
66 views

Roll a 6-sided fair die until a 6 appears. Let X = the number of 1's that are rolled. Find Var(X).

Let X = the number of 1's that are rolled. Find E[X] and Var(X). I can't seem to calculate Var(X). I've calculated E[X] = 1. I let R = the number of non-6 rolls, and I let Y = the number of rolls ...
0
votes
0answers
24 views

Computing variance of a proportion

I had a question regarding this paper. In page 3, they show the way to estimate $\pi$ as $$ \pi = \frac{\lambda + p - 1}{2p - 1} $$ and then they proceed to compute the variance as $$ Var(\pi) = ...
0
votes
1answer
31 views

Bounded function of geometric random variable

if X~ Geometric(p), with q=1-p, then show that for any bounded function f with f(0)=0, we have E(f(x)-qf(x)+1)]=0. Our professor asked us to try solving this problem as a good practice but I have no ...
1
vote
0answers
46 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
1
vote
2answers
62 views

Maximum likelihood estimator?

I am looking at some questions from Mods 2010 and I can't figure this one out. I think my problem is technical... We have a sample (L1,R1), ...,(Ln,Rn) with Lj and Rj normally distributed independent ...
2
votes
1answer
44 views

Calculating joint MGF

This is an end-of-chapter question from a Korean textbook, and unfortunately it only has solutions to the even-numbered q's, so I'm seeking for some hints or tips to work out this particular joint ...
0
votes
1answer
25 views

Mean Square Estimate problem

I have to find $\textbf{s}_{MS}$ given $\textbf{r} = h\textbf{s}+\textbf{n}$ where $h$ is a Bernoulli random variable with $Pr(h=1)=Pr(h=0) = 1/2$ and $\textbf{s}$ and $\textbf{n}$ are independent ...
1
vote
0answers
46 views

If $X$ is a random variable, under which conditions is $g(X)$ also a r.v.?

In many instances, functions of random variables appear, and we usually treat them as random variables also. In the 3d edition, pp. 85-86, of this well-known book (now in its 4th edition), we find the ...
3
votes
0answers
48 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
0
votes
0answers
9 views

Estimating variance from the sequence

Suppose that we have $\{X_n\}\to X\sim N(0,\Omega)$ where $X_n$ can be obtained from observations. My problem is to estimate $\Omega$ consistently. If $var X_n$ converges to a "finite" matrix, then ...
0
votes
2answers
172 views

Are any linear combination of normal random variables, normally distributed?

It is easy to show that if we have n independent normally distributed random variables, then a linear combination fo them ar normally distributed. It is also said that if (x1,x2,..,xn) is ...
1
vote
2answers
19 views

Show that $Cov(X,Y) \geq -23$

if $X,Y$ are two random variables and: $Var(X) = Var( Y) = 23$ how can i show that $Cov(X,Y)\geq -23$ can someone give me some hints on how to show it?(not an answer) i know that $Cov(X,Y) = E(XY) ...
1
vote
0answers
19 views

What is the dot product of two randomly generated 0-mean unit-vectors?

Given pairs of random 0-mean unit vectors, what kind of distribution is generated by their dot products? Judging from a number of results I've generated myself, the distributions appear to be ...
2
votes
1answer
55 views

What's the distribution of the exponential of uniformly distributed variable?

I want to know the distribution of $z = \exp(j\varphi)$, with $\varphi \sim \mathcal{U}[-\pi;+\pi]$. From the book "Probability, Random Variables and Stochastic Processes" by Papoulis and Pillai I ...
0
votes
1answer
32 views

Covariance Matrix of zero mean complex vector

$$\textbf{f}=[f_1, f_2, f_3];\quad \textbf{g}=[g_1, g_2, g_3] $$ $f_1,f_2,f_3,g_1,g_2,g_3$ are all independent identically distributed zero mean complex random variable. h = elementwise wise ...
2
votes
3answers
51 views

Probability computation, tossing two dice

I have some ideas on how to solve the problem, but simulations do not support my analytical results :) Toss two dice and sum their value and write it down: Denote by $X_n$ the result at $n$-th toss. ...
0
votes
2answers
71 views

Adding two normal distribution

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. And Suppose that $Z \sim N(1, 2^2)$ and is independent of all $X_i$. Define $Z_i = Z + X_i$ for $i = 1, ...
0
votes
2answers
50 views

Central Limit Theorem Application on Multivariate Normal

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. What is the distribution of $\overline{X} = \frac{1}{3}(X_1+X_2+X_3)$? I don't quite understand how to ...
1
vote
1answer
24 views

measured variables with almost zero variance

I am interested in knowing some examples for measured variables, which show almost zero variance. Could anyone list me some examples? I am not sure, but perhaps the measured variable "speed" of a ...
2
votes
2answers
66 views

Given X and Y are correlated and Y and Z are correlated what is the range of correlation between X and Z?

How can I calculate the range of correlation of two variables X and Z given I have the correlations of X and Y, and Y and Z? I've found a few resources around, namely this, but I'd like a research ...
0
votes
1answer
35 views

Multiplying a vector of independant gaussian r.v. by an orthogonal matrix gives independant r.v.

I found in a proof (about $\chi^2$ and Student laws) : Let $$\begin{pmatrix}V_1 \\ V_2 \\ ... \\ V_n \end{pmatrix} = A \begin{pmatrix}Z_1 \\ Z_2 \\ ... \\ Z_n \end{pmatrix}$$ Since $Z_i$ ...
1
vote
1answer
35 views

How does one model independent trials in statistics.

In my probability class, we covered the proof of the following result, known as the "strong law of large numbers": Theorem. Let $(\Omega,\mathscr F,P)$ be a probability space, $\{X_n\}_{n\in\mathbb ...
0
votes
1answer
46 views

Sampling random numbers with a certain condition.

I want to randomly sample three variables that are conditioned by $$x_1 \le x_2 \le x_3$$ and $x_1\in [0,\, \ell]$, $x_2\in [0,\, \ell-\ell_1]$ and $x_3 \in [0, \,\ell-\ell_1-\ell_2]$. I have only ...
0
votes
2answers
55 views

what is meaning of “independent values of a random variable”?

I need some help with basic statistics terminology. Could someone please explain in layman terms the meaning of "independent values" regarding a random variable? Perhaps a six-sided die (with sides ...
0
votes
0answers
31 views

If $P_{\theta_0} (X_i \leq x) \leq P_{\theta_1}(X_i \leq x)$, then $P_{\theta_0} (\sum X_i \leq x) \leq P_{\theta_1}(\sum X_i \leq x)$

If $P_{\theta_0} (X_i \leq x) \leq P_{\theta_1}(X_i \leq x)$. Is it true that: $P_{\theta_0} (\sum_{i=1}^nX_i \leq x) \leq P_{\theta_1}(\sum_{i=1}^n X_i \leq x)$ I'm doing a long ...
1
vote
2answers
23 views

Is this definition of a quantile proper?

I need to find a proper definition of a quantile. It says: a p-th quantile $x_p$ is a number, that satisfies the following conditions: $$ 0<p<1 $$ and $$ P(X \le x_{p}) \ge p $$ and $$ P(X \ge ...
1
vote
2answers
68 views

A Question About Sampling

I have a large population of size n from an unknown continuous random variable X, and I do not know the underlying distribution of X. Suppose that I know the minimum sample size b required to ...
2
votes
0answers
27 views

Evaluate spatial variation of density-like scalar

Apologies if this has been asked previously, but I'm not totally sure of the best way to pose the question. Background I'm evaluating the variation of a spatially varying scalar field $p$ ...
3
votes
1answer
46 views

The number of nuts in a package of “Premium Cashews” is normally distributed with a mean of 433 and a standard deviation of 6 nuts.

The number of nuts in a package of "Premium Cashews" is normally distributed with a mean of 433 and a standard deviation of 6 nuts. Packages with fewer than 420 nuts or more than 445 nuts will be ...
1
vote
2answers
44 views

Finding probability that the function of a random variable is less than another random variable

X and Y are random variables, whose joint density function is $f(x,y)$ for $\infty<x<\infty$ and $\infty<y<\infty$. I am trying to find $P[X^2<Y]$. Here's how I plan on solving the ...
2
votes
4answers
226 views

Expected value of number of cards drawn from a deck to get 5 spades.

The question: What is the expected number of cards required to be drawn in order to draw 5 spades. What I have: Let $X=X_1+\cdots+X_{43}$ (43 because we're examining the case when 4 spades have been ...
1
vote
1answer
37 views

Finding probability that two discrete random variables are equal

I have two random and independent variables, X and Y, whose probability distribution functions are given by $f(x) = (1/10)(9/10)^x$ and $f (y) = (1/5)(4/5)^y$, for X = 0, 1, 2, ... and Y = 0, 1, ...
0
votes
1answer
95 views

Combining random variables

On combining random variables and their means, this article states: Suppose you have two variables: $X$ with a mean of $μ_{x}$ and $Y$ with a mean of $μ_{y}$. The mean of the sum of these ...