Tagged Questions

A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).

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40 views

Exponential random variable

The time it takes a printer to print a job is an exponential random variable with mean of 10 seconds. You send a job to the printer at 9:00 am, and it appears to be the fourth in line. What is the ...
2
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0answers
32 views

Sum of Bernoulli random variables

I need some help with a homework assignment. The question I'm given is: "Suppose that $X_1, X_2,..., X_n, W$ are independent random variables such that $X_i\sim Bin(1,0.4)$ and $P(W=i)=1/n$ for $i=1, ...
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1answer
25 views

How to bound a probability with Chernoff's inequality?

In my class, we were given Chernoff's inequality as $$P(X\le -t) \le e^{(-(\lambda*t - \log( E(e^{-\lambda*x}))))}$$ $$P(X\ge -t) \le e^{(-(\lambda*t - \log( E(e^{\lambda*x}))))}$$ It says that to ...
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2answers
72 views

Expectation of $(X + Y)^2$ where $X$ and $Y$ are independent Poisson random variables

I would really appreciate anyone's help with this problem: (let $E$ denote expectation) Suppose $X$ and $Y$ are independent Poisson random variables, each with mean $1$. Find: $E[(X + ...
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1answer
19 views

Sum of dependent R.V

I have two random variables whose PDF are parameterized by an unknown constant as follows: P(A;d) P(B;d) apparently, these two are not independent, so to find P(A+B;d) one cannot use convolution. ...
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1answer
39 views

sum of correlated random sample

Suppose I have 1000 draws each of two random variables X and Y. If I wanted to sample the sum of these variables, I would simply calculate 1000 samples, i.e. $$ S_{i}=X_{i}+Y_{i}, i=1,2,…,1000 $$ ...
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0answers
11 views

G and R matrices in mixed model and model selection

I have data in which the plants were subjected to four conditions and measured weekly for a month. I would like to incorporate "plot" as a random factor into my linear mixed model using SPSS. I am ...
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0answers
29 views

Independence of functions of Random variables [closed]

I am working with two RV's X and Y X is distributed normally with mean=x and variance=x^2 Y is distributed uniform (0,1) I know this... ...
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1answer
42 views

Covariance of a compound distribution

I am trying to find the covariance of a compound distribution. Given $X=x$, where $X \sim \mathrm{Uniform}(0,1)$, $Y$ is (conditionally) normally distributed with mean $x$ and variance $x^2$. I ...
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1answer
29 views

Bayesian Network

I am preparing for midterm exam and need to know what is the step by step solution to this question? Answer is shown in red. Also any external related link is very much appreciated.
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2answers
89 views

How to prove dependence of random variables

I need to solve the following problem. Let $X$ be a normal random variable with mean $\mu$ and standard deviation $\sigma$ and let $I$, independent of $X$, be such that $\mathbb{P}(I = 2) = ...
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0answers
15 views

Convergence of Random Variables meaning

What is the intuitive explanation of convergence of random variables? what is meant by saying that a sequence of random variables CONVERGE?
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1answer
26 views

Sequence of Random Variables

I am confused about how to approach sequence of random variables that are not identically distributed. For example, consider a sequence $X_1, X_2, \dots, X_n$ with the pdf: $$ f(X_n)= \begin{cases} ...
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0answers
21 views

Sampling from a distribution with a margin of error

A survey of a population is taken using sampling. It is determined that 70% prefer option A and 30% prefer option B with a margin of error being 5%. Normally when simulating the process with the ...
1
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0answers
23 views

Number of trials to observe all values of a uniform discrete random variable X with a probability of at least 1-q?

Let X take on p values with equal probability. If n trials are to be conducted to ensure that the probability of not observing any of these p values is less than or equal to q, what is the value of ...
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0answers
18 views

Linear Combination of Random Normal Variables

In order to prove that the linear combination of two independent normal distributions(say Z=X+Y) is normal, i am using their MGFs to show that the linear combination also has a similar mgf. This works ...
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0answers
39 views

Distribution for $Y = \sqrt{X_1^2 + X_2^2}$, when $X_1, X_2$ are dependent and normally distributed with different variance? [duplicate]

Is there a closed form distribution for the transformation given by $Y = \sqrt{X_1^2 + X_2^2}$, when $X_1, X_2$ are jointly normal but dependent random variables with different variance? OBS: I know, ...
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1answer
60 views

Combination of Random Variables Conditional Probability

If A and B are independent discrete random variables and C = A+B, then how should one compute the pmf of P(A|C)? For example, let X be the result from a coin toss(1 or 0 for H and T) and Y be the ...
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1answer
34 views

Probability distribution, unfair coin

An unfair coin which has 0.35 probability to result head is tossed four times. Build and represent graphically the probability distribution and the cumulative ...
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0answers
32 views

Are matrix Fisher r.v.s closed under multiplication?

With appropriate parameters, a matrix Fisher distribution provides a distribution over SO(3) (i.e. over rotations in $R^3$). See this MathOverflow post for a few notes describing the distribution. ...
1
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1answer
36 views

Can an a.s. (almost surely) finite random variable be a.s. UNbounded?

I thought that if a random variable, $\eta^2$, is assumed to be a.s. finite, then $\eta^2$ must be a.s. bounded. In the Martingale Central Limit Theorem in Hall & Heyde, they assume this: "let ...
2
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1answer
70 views

Moment Generating Function of a nonlinear transformation of an exponential random variable

Let $\tau$ be an exponential random variable, with parameter $\lambda$. Let $$ V = \delta^\tau $$ where $0 < \delta <1$. Sorry if this notation seems strange, but it is what I am using, I ...
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31 views

How to find the significantly different groups in Random Effects

I am using a mixed effects model as created here (using dummy data for now) in this R script ...
4
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2answers
120 views

Calculate $\mathbb{E}[Z_i]$ where $Z_i = \min(X_i, Y_{i-1})$, $X \sim Beta(\alpha,1)$

Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim \mathrm{Beta}(\alpha,1)$. Let $Z_i = \min(X_i,Y_{i-1})$, $\forall i >1$. I would like to ...
2
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1answer
46 views

Sum of Random Variables

As part of my statistical mechanics class, I'm trying to go through Kardar's statistical physics of particles and I'm having trouble with this one line: Consider the sum $X=\displaystyle ...
2
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2answers
88 views

Skewness of a random variable that have zero variance and zero third central moment

If I have a random variable $x$, and the only information I know about it are: $$ m_1=E[x]=c, \mu_2=var(x)=0, \mu_3=E[(x-m_1)^3]=0$$ Can I conclude that the function distribution is symmetric about c? ...
2
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1answer
37 views

How is this Negative Binomial Random variable used to solve this problem?

I was looking at the solution to this problem below and I don't understand how they used a negative binomial R.V. to solve the problem. A research study is concerned with the side effects of a new ...
4
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1answer
174 views

Do mean, variance and median exist for a continuous random variable with continuous PDF over the real axis and a well defined CDF?

For a continuous random variable with continuous PDF over the real axis and well defined CDF, are the mean, variance, and median always well defined? Mean and variance do not always exist, e.g. for a ...
0
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1answer
266 views

Expected Value of Random Variable

I'm trying to find the expected value of a random variable $t_i$ which is the solution of $$\epsilon_i=\mu(t_i-t_{i-1})-\sum^{i-1}_{k=1}\frac{\alpha}{\beta}\left(1-e^{-\beta(t_i-t_k)}\right)$$ ...
2
votes
1answer
38 views

Does the average of the square roots of random variables mean anything?

I recently made a plot for work that used a signed square-root scale on the $y$ axis, for visual clarity. The $y$ observations are impulse response functions (IRF) of vector autoregressions computed ...
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0answers
10 views

probability life similarities in a population cross section

I have just read up on Bouchard's Minnesota Twins study that turned up some amazing identical similarities in the lives of some identical twins reared apart. Both had same jobs, married and divorced ...
0
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1answer
28 views

Random variable variance

I have the model $y_i=\beta_1+\beta_2 X_i+ u_i$ where $u_i\sim\text{iid } N(0,\sigma^2)$. I estimate $\beta_1$ and $\beta_2$ by drawing a straight line between the first $(x_1,y_1)$ and last dot ...
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0answers
126 views

Applying a variance-stabilizing transform to a fitted function (rather than data)

Outline I'm working with data corrupted by a mixed Poisson-Gaussian noise model (for example with images gathered in astronomy or electron microscopy), and have been using the generalized Anscombe ...
2
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0answers
19 views

Estimating number of intersections between point and polygons

I have a 2D plane (a large rectangle) with a finite size in the x and y direction, which is the field of my problem. The field is covered by $n$ smaller rectangles that are located randomly within ...
4
votes
3answers
65 views

What is the correct notation if two random variables belong to the same distribution?

I want to explain that both the real and imaginary part of a complex variable follow a zero-mean complex Gaussian distribution. How can I write that? For one variable I'd write $a \sim ...
1
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0answers
44 views

probability that a variable is ONE OF 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 non-random variable which can vary across $i$ and $X_i$ is a random variable with Pareto Type I distribution. ...
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3answers
126 views

Biased coin toss simulation — which random generator is most appropriate?

I have a doubt whether the uniformly distributed random generator is the most appropriate to use for simulating biased coin toss. ...
3
votes
4answers
43 views

Conditional variance - $Var(X + U | X) = Var(U)$?

I am wondering if the following equality holds - $Var(X + U | X) = Var(U)$? where $X$ and $U$ are two independent random variables? It seems can we say $Var(X + U | X) = Var(X|X) + Var(U|X) = ...
3
votes
1answer
31 views

Notation: Deterministic Variable, Random Variable, Realization of Random Variable, Function

Is there a standard notation for distinguishing between the following: Random Variables Realizations of Random Variables Deterministic Variables (not random) Functions I am familiar with using ...
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0answers
36 views

How to apply the random statement in my model?

i am trying to model the following data in SAS using proc glimmix and i would like feedback if i have modelled my data correctly. My data consist of individual chickens (ID) who are grouped by ...
2
votes
2answers
57 views

Probability generating function for negative values of random variables?

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer ...
1
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1answer
21 views

Laplace transform and density

It is true that the Laplace transform of a (positive) random variable characterises that random variable, just like its density? ($L_X(z) = E(exp(-Xz))$)
2
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0answers
39 views

Largest eigenvalue of this random 2x2 matrix

Consider four random complex vectors $\mu_i$ of length $K$ whose entries are drawn from the complex normal distribution $\mathcal{CN}(\mathbf{0},\mathbb{1})$ centered in zero and of unit variance. ...
0
votes
0answers
20 views

Measure of probability of obtaining a certain $\chi^2$ given the number of trials

I am calculating an equation with several random components a large number of times, and in order to select my final answer I choose that with the minimum value for $\chi^2$. I am hoping to find some ...
3
votes
1answer
97 views

Squared random variable $X^2$ vs $X\times X$

As I understand a random variable represents all possible outcomes of an experiment with their associated probabilities. Why $X^2$ is understood as squaring outcomes of experiments instead of as ...
2
votes
1answer
119 views

If a random variable V is independent of two independent random variables X and Y, how to prove that V is independent of X + Y?

This is question 3.8.4 of An Introduction to Mathematical Statistics and Its Applications, 5th Edition, by Larsen and Marx. This is not homework for a class I am taking now, but might someday be for ...
2
votes
1answer
46 views

correlation of two sums of random variables

Imagine two random variables $X$ and $Y$ which are correlated with $\rho = 1$. Both have a mean of $100$ and a standard deviation of $40$. Two other random variables $U$ and $V$ are correlated at ...
2
votes
1answer
29 views

Dirichlet sample by normalising Gamma RVs

I know that if you sample $K$ random variables $(X_1, X_2, \dots, X_K)$ from Gamma distributions using shape parameters $(\alpha_1, \alpha_2, \dots \alpha_K)$ and a scale parameter $\theta = 1$ such ...
2
votes
1answer
50 views

Rectifier function of random variable

Let $X$ be a random variable with distribution $F_X$ and density $f_X$. Define $$g(x) = \left\{ \begin{array}{lr} x & : x \ge 0\\ 0 & : x < 0 \end{array} \right\}$$ and let ...
1
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1answer
77 views

Probability function of three Random Variables multiplied, solidifying intuition

I ran across this exercise: Let $T$ be a random variable distributed as a $\text{Bernoulli}(p)$, $U$ be a random variable distributed as a $\text{Bernoulli}(q)$ and $W$ be a random variable ...