# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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) = ... 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 ... 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. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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) ... 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 ... 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 ... 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] $$f_1,f_2,f_3,g_1,g_2,g_3 are all independent identically distributed zero mean complex random variable. h = elementwise wise ... 3answers 51 views ### Probability computation, tossing two dice I have some ideas on how to solve the problem, but simulations do not support my analytical results :) Toss two dice and sum their value and write it down: Denote by X_n the result at n-th toss. ... 2answers 71 views ### Adding two normal distribution Suppose that X_1, X_2, X_3 are i.i.d. normal random variables with mean 0 and variance 1. And Suppose that Z \sim N(1, 2^2) and is independent of all X_i. Define Z_i = Z + X_i for i = 1, ... 2answers 50 views ### Central Limit Theorem Application on Multivariate Normal Suppose that X_1, X_2, X_3 are i.i.d. normal random variables with mean 0 and variance 1. What is the distribution of \overline{X} = \frac{1}{3}(X_1+X_2+X_3)? I don't quite understand how to ... 1answer 24 views ### measured variables with almost zero variance I am interested in knowing some examples for measured variables, which show almost zero variance. Could anyone list me some examples? I am not sure, but perhaps the measured variable "speed" of a ... 2answers 66 views ### Given X and Y are correlated and Y and Z are correlated what is the range of correlation between X and Z? How can I calculate the range of correlation of two variables X and Z given I have the correlations of X and Y, and Y and Z? I've found a few resources around, namely this, but I'd like a research ... 1answer 35 views ### Multiplying a vector of independant gaussian r.v. by an orthogonal matrix gives independant r.v. I found in a proof (about \chi^2 and Student laws) : Let$$\begin{pmatrix}V_1 \\ V_2 \\ ... \\ V_n \end{pmatrix} = A \begin{pmatrix}Z_1 \\ Z_2 \\ ... \\ Z_n \end{pmatrix}$$Since Z_i ... 1answer 35 views ### How does one model independent trials in statistics. In my probability class, we covered the proof of the following result, known as the "strong law of large numbers": Theorem. Let (\Omega,\mathscr F,P) be a probability space, \{X_n\}_{n\in\mathbb ... 1answer 46 views ### Sampling random numbers with a certain condition. I want to randomly sample three variables that are conditioned by$$x_1 \le x_2 \le x_3$$and x_1\in [0,\, \ell], x_2\in [0,\, \ell-\ell_1] and x_3 \in [0, \,\ell-\ell_1-\ell_2]. I have only ... 2answers 55 views ### what is meaning of “independent values of a random variable”? I need some help with basic statistics terminology. Could someone please explain in layman terms the meaning of "independent values" regarding a random variable? Perhaps a six-sided die (with sides ... 0answers 31 views ### If P_{\theta_0} (X_i \leq x) \leq P_{\theta_1}(X_i \leq x), then P_{\theta_0} (\sum X_i \leq x) \leq P_{\theta_1}(\sum X_i \leq x) If P_{\theta_0} (X_i \leq x) \leq P_{\theta_1}(X_i \leq x). Is it true that: P_{\theta_0} (\sum_{i=1}^nX_i \leq x) \leq P_{\theta_1}(\sum_{i=1}^n X_i \leq x) I'm doing a long ... 2answers 23 views ### Is this definition of a quantile proper? I need to find a proper definition of a quantile. It says: a p-th quantile x_p is a number, that satisfies the following conditions:$$ 0<p<1 $$and$$ P(X \le x_{p}) \ge p $$and$$ P(X \ge ...
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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 ...
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### 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 $p$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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, ...
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 ...