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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79 views

Distribution of the quotient of two gamma random variables with different rate parameters?

I have a question about how to derive the distribution of the quotient of two random gamma variables drawn from two different Gamma distributions with the same shape, but different rates. For example, ...
2
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2answers
115 views

Is a random variable Bernoulli? Is a proof available?

Suppose a die is tossed twelve times and each outcome is represented by a random variable $X_{i}$. Further define $Y_{i}$ for $i=2,...,12$ to take the value $1$ if $X_i=X_{i-1}$ and $0$ otherwise. ...
7
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1answer
117 views

If $X_1,X_2$ are independent beta then show $\sqrt{X_1X_2}$ is also beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities ...
2
votes
1answer
17 views

Correlation of one random variable and a binary variable dependent on it

I create one normal random variable using =NORM.S.INV(RAND()) in Excel and a binary variable =IF(B2>=0,1,0), where B2 is this random variable. If I continously resample, I find that their correlation ...
2
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1answer
48 views

Confusion about the sample distribution.. Can you please enlighten me?

I thought that the sample distribution was an approximation of the distribution of the underlying phenomenon. But then the book says: We will denote the sample size by $n$ ($n \le N$) and the ...
2
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2answers
94 views

Random variables independence

I need to check if $Z$ and $W$ are dependent or not. $X,Y \sim \mathrm{Exp}(2)$ Then I define: $Z=X-Y \ \text{,}\ \ W=X+Y$. Now How I can check that $Z$ and $W$ are dependent or not ? I know ...
2
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1answer
61 views

Example of sample $X_1,X_2,\ldots,X_n$

In the book Statistical Inference by George Casella, it is written that An experimenter uses the information in a sample $X_1,X_2,\ldots,X_n$ to make inferences about an unknown parameter $\theta$. ...
2
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1answer
33 views

Chi-squared distribution and dependence

We know that for a group of independent random variables $\{X_i\}_{i=1}^n$ s.t each $X_i$ is distributed as a chi-squared distribution with degree of freedom one ($\chi_1^2$), the sum of these random ...
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0answers
13 views

Why is it inappropriate to use a post hoc test on a random effect?

This is a bit more conceptual, but I know its incorrect to run a post hoc test on a random effect, though I'm not entirely sure as to why. I figure it has to do with the selection of random variables, ...
2
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0answers
32 views

Finding MLE with ordered statistics?

Let Y1 < Y2 < ... < Yn be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval: $$[\theta - \rho, \theta + \rho]$$ ...
1
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1answer
35 views

CDF of the function of a random variable

I haven't been able to find useful information on this. I was just wondering what would be the distribution of a function of a random variable. For example, what would be the distribution of the ...
0
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3answers
101 views

What is the difference between variable and random variable?

I know that "variable" means "values which vary." In a simple linear regression model : $$Y=\beta_0+\beta_1X+\epsilon$$ $X$ is variable that is the values of $X$ vary. Why is $X$ not a random ...
0
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2answers
101 views

Probability of finding a point in the unit circle?

Consider the experiment where a pair of numbers (x,y) is chosen at random in the unit square; that is, x and y are uniform (0,1) random variables. What is the probability of (x,y) lying within the ...
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0answers
28 views

Is sampling according to several i.i.d. random variables or to just one of them equivalent

Suppose we have n+1 random variables i.i.d. distributed $X_0,X_1,...X_{n}$. Is it the same operation if I generate n samples according to $X_0$ $\{S^0_1,S^0_2..,S^0_n\}$, or if I take from each of the ...
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0answers
11 views

How to find anti-correlated subsequences in correlated time series?

Say I have two time series $X_t$ and $Y_t$ (with $ 1 \leq t \leq N$), which have a high positive Pearson correlation. Say I also have reason to believe there are subsequences $X_{tj}, Y_{tj}$ (where, ...
6
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1answer
92 views

Parameters vs latent variables

I have asked about this before and have really been struggling with identifying what makes a model parameter and what makes it a latent variable. So looking at various threads on this topic on this ...
3
votes
1answer
59 views

PDF of function of X

I'm learning about functions of random variables and am trying to work out an example I made up. If $y = \sin(x)$ and $x$ has domain $[0, 4\pi]$, is the following the correct expression for the pdf of ...
2
votes
1answer
43 views

On an implication of the memoryless property of the exponential random variable

I know that if we take $X \sim Exp(k)$ then we have this property: $$P(X \ge s + t | X \ge s) = P(X \ge t)$$ But why does this imply that $X | X > x$ has the same distribution of $X$ only ...
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0answers
29 views

“negligible” random variables

I would like to ask if the concept of 'negligible' random variables exist. For example, if $X$ and $Y$ are random variables, not necessarily independent, then the sum $X+Y$ is, for all intents and ...
0
votes
1answer
25 views

Variance calculation RELU function (deep learning)

weight initialization is important for modern deep learning. To understand [1,2], I would like to understand the following: $$ E[x^2] = 0.5 Var[y], $$ where $x= max(0,y)$, $E[.]$ is the expectation, ...
2
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1answer
43 views

Distribution of the product of a gamma random variable and a beta random variable

When you multiply a gamma random variable with a beta random variable, you should get a gamma random variable. I'm having a little trouble showing this, though. I figure I'm forgetting some clever ...
3
votes
1answer
107 views

correlation of r1 and r2 is x. Probability of r1 > r2?

Just a quick probability interview question. If the correlation of two variables r1, r2 is x. What's the probability that a sample of r1 is greater than a sample of r2? let me update the question. ...
3
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1answer
54 views

Continuous random variable and median problem

The definition of median, $m$, for a continuous random variable $X$ is $$P(X\leq m)=P(X\geq m)=\int_{-\infty}^m f(x)\text{d}x=\int_{m}^{\infty}f(x)\text{d}x=\frac{1}{2}$$ where $f(x)$ is the pdf of ...
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0answers
26 views

Notation for a random vector whose length depends on another random variable?

I have the following process that I'm trying to describe with random variables. First, I have a random variable $X$ that takes on values drawn from a Poisson distribution with parameter ...
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1answer
59 views

Sum of iid random variables

Let $X_1, X_2,...,X_n$ be iid random variables. Let $Z_1, Z_2, Z_3$ be defined as $X_1, X_1+X_2, X_1+X_2+X_3$ respectively. Are $Z_1, Z_2$ and $Z_3$ also iid's? The question is based on renewal ...
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3answers
56 views

Intuition of pdf of a continuous random variable [duplicate]

What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if ...
2
votes
1answer
13 views

Using variables that are only available for part of the data-set in a classification model

I have Data X1, X2, and y. X1 has the same variables as X2, + some extra variables that X2 does not have. I want to use the data X2 to predict binary variable y. I suspect the extra variables In ...
2
votes
1answer
58 views

Mixed effect modelling with multiple, nested random variable

Goal: comparing pitch (Hz) on three types of words Dependent variable: Hz Fixed predictor variable: word-type, points (measurements taken from five points on each token, to capture Hz change within ...
4
votes
1answer
44 views

Linear transformation of a random variable by a tall rectangular matrix

Let's say we have a random vector $\vec{X} \in \mathbb{R}^n$, drawn from a distribution with probability density function $f_\vec{X}(\vec{x})$. If we linearly transform it by a full-rank $n \times n$ ...
7
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2answers
381 views

pdf of a product of two independent Uniform random variables

Let $X$ ~ $U(0,2)$ and $Y$ ~ $U(-10,10)$ be two independent random variables with the given distributions. What is the distribution of $V=XY$? I have tried convolution, knowing that $$h(v) = ...
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0answers
14 views

How to show via Delta method that the Linear Taylor series expansion of a normal random vector results in NORMAL DISTRIBUTION [duplicate]

How can it be proved using the delta method that the Linear Taylor series expansion of a normal random vector containing independent but NOT identically distributed elements results in a random ...
3
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0answers
32 views

If X is independent of Y given Z, is X/Z independent of Y?

When is the following true? If $X$ is independent of $Y$ given $Z$, $\frac{X}{Z}$ is independent of Y.
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1answer
58 views

How to find the distribution of a function of multiple, not necessarily independent, random variables?

If $Y$ is a random variable defined as $Y=g(X_1,X_2)$, where $X_1$ and $X_2$ are two different random variables whose distributions are known (say with pdf's $f_{X_1}$ and $f_{X_2}$), how do we find ...
0
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1answer
47 views

Generating random simulations of events [closed]

Whole-edited to make it more simple. Let's assume we have a concrete event: A baseball player's batting average is 0.32. I want to find a random number X, that ...
6
votes
2answers
134 views

Linear combination of discrete variables $T_i$ with $P(T_i=1)=P(T_i=-1)=1/2$

Let $T_1,...,T_n$ be iid with a Rademacher distribution; i.e., $P(T_i=1)=P(T_i=-1)=1/2$; and let $w = (w_1,...,w_n) \in \mathbb{R}^n$ without further constraints on $w$. Is there a way to compute ...
3
votes
3answers
188 views

The probability of a random variable being larger than a sequence of random values

Suppose we have a fixed, known, $n$, and each $x_1 \ldots x_n$ is a random number generated uniformly over $[0,1]$. What is the probability that $x_n$ is the largest value in the sequence?
2
votes
1answer
45 views

Can the population be smaller than the number of possible outcomes?

Say a random variable $X$ returns one of the values $1, 2, ..., 6$. Conceptually, does it make sense to speak of a population, where each individual element is just $1$? In other words, does the ...
4
votes
1answer
304 views

Why is this random variable both continuous and discrete?

The waiting time, $W$, of a traveler queuing at a taxi rank is distributed according to the cumulative distribution function, $G(w)$, defined by: $$G(w) = \begin{cases} 0 & \text{ for } ...
0
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0answers
83 views

tobit model please write down

Let $y$ be response variable taking on values $[0,1]$ , where $P(y=0)=0$ and $P(y=1)>0$ . An example is, for a random sample of markets indexed by i, $y_i$ is the share of the largest firm; by ...
1
vote
1answer
30 views

Expected value of the inverse of a random variable

Let $X$ be a random variable. $X$ can take the value 1 with probability $p$, and the value 2 with probability $1-p$. Can we write $E[\frac{1}{X}] = \frac{1}{E[X]}$? (note that $E[X] \neq 0$) Thank ...
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vote
0answers
19 views

How to decide which of two distributions a sample is from?

I have two random distributions. How can I get a quantitative estimate of the likelyhood that a particular sample was taken from one or the other of the distributions? To be more precise: Let $I = ...
1
vote
1answer
43 views

Exclude Some samples for calculating CDF

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and ...
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1answer
46 views

What is the expected number of coin flips, if you stop when the first coin flip is the same as the last?

In order to calculate the $\text{E}[X]$ where $X$ is the number of total coin flips, this is the approach I took: The probabilities are: $Pr(H) = p$ $Pr(T) = (1-p)$ Define indicator random ...
2
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2answers
63 views

Weakly correlated Random variables

If $N$ random variables are identically distributed but weakly correlated, in what condition we can approximate them as independent identically distributed (iid) ? I saw an old paper where based on ...
0
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1answer
19 views

How to compute $\mathbb{P}(A|B)$ of two independent RVs? [duplicate]

There are two independent RVs $X \sim \mathcal{U}(-1,5)$ and $Y \sim \mathcal U(-5,5)$. Let $A = \{ X \ge Y \land Y \ge -1 \land Y \le 1\}$ and $B = \{ X \le 1\}$. What is $\mathbb P (A | B)$? My ...
3
votes
1answer
80 views

Conditional expectation of $\mathbb{E}(X - Y | (X, Y)\in\mathcal{A})$

Given two independent random variables $X \sim \mathcal{U}[-1,5]$ and $Y \sim \mathcal{U}[-5,5]$, what is $$\mathbb{E}\{Y - X | X \le 1, Y > X, Y \in [-1,1] \}\,?$$ I managed to compute the ...
0
votes
1answer
57 views

Multiplication of two random distribution

I am trying to find the resulting PDF , when two random functions are multiplied. First function obeys normal distribution and second function obeys cauchy distribution. Can anybody tell me how to ...
0
votes
1answer
116 views

What is the correlation between X and X+Y?

If $X$ and $Y$ are two random variables, how do I calculate the correlation of $X$ and $X+Y$ in terms of $\rho$, $σ_x^2$ and $σ_y^2$ given that the $\text{Variance}(X)= σ_x^2$ and ...
3
votes
2answers
108 views

What is the meaning of the conditional $y|b$

I think I'm confused about a very simple thing. When we say that some variable is distributed as a Poisson distribution and we write $y \sim \text{Pois}(\lambda)$, is this the same that saying ...
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votes
2answers
88 views

Example of a random variable that is not iid

What is an example of a random variable that is not i.i.d? The usual ones (coin flips, rolling of a dice) are all i.i.d, so I am trying to understand what is an example of a random variable that is ...