# Questions tagged [random-variable]

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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### Generation of random variables via composition and inversion

What are the main pros and cons of each method and when to use each one? Law [2007] mentions that: "Again, the reader is encouraged to develop the inverse-transform method for generating a ...
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### Is it possible that marginally independent random variables are conditionally dependent?

Suppose that $X,Y$ and $Z$ are random variables. If $X$ is independent of $Z$ and $Y$ is independent of $Z$, is it possible that $X$ is dependent on $Z$ given $Y$ and $Y$ is dependent on $Z$ given $X$?...
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### Is a “random variable” still a random variable if it is predictable?

If X in P(X) is considered a random variable because it varies among the occurrence the particular scenario which it may occur (such rolling a number on a die), can we still call X a “random variable” ...
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### finding probability distribution of sum of 2 random variables

I have a probabiliy distribution $$p(x) = \begin{cases}e^{-x} & x\geq0\\ 0 & x<0\end{cases}$$ I need to find the probability distribution for $Z=X+Y$ where X and Y are from the above ...
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### Truncated expectation of sum of independent random variables

Take three random variables $X$, $Y$, $Z$ s.t. $E[X]>0$, $E[Y|X]=0$, $Z = X+Y$. What can I say about $E[x| x> k]$ vs. $E[z| z>k]$ where $k>0$? Intuitively, the latter should be bigger ...
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### Finding standard deviation of formula with three variables

Suppose we have the formula $Y = \frac{2WV^{2}}{\pi D^2}\hspace{1cm}$(1) where: w = weight variable, v = velocity variable, d = diameter variable we want to find $SD(Y)$ the solution which was ...
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### Why is the explanatory variable non-stochastic or fixed in repeated samples?

I am studying econometrics. I have been learning about deriving the variance for the OLS slope statistic in a simple linear regression model. Why is the explanatory variable considered to be non-...
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### Find PDF of Z=X/(Y+c), c a constant and given independence of X and Y and the PDF of X and the PDF of Y

I want to find the PDF of $Z=X/(Y+c)$ where $c$ is a constant and $X,Y$ are two independent random variables. The PDFs of $X$ and $Y$ are supposed to be given. I would like to have a general form ...
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### Z-scores and Probability

If $H$ is a normally distributed random variable with expected value 1.52 with standard deviation of 0.74. What is the probability that p(H=1.52)? Maybe I'm overthinking this but would you simply find ...
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### Covariance matrix as a linear transformation

I am trying to understand the general relationship between the covariance between two random variables and linear transformations. For example, consider the figure here: https://en.wikipedia.org/wiki/...
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### Good measures to quantify the effect of outliers

I have two random quantities linked to a set of N random variables. Increasing N The two random quantities should gradually become similar (meaning that the 2 distributions narrows around the same ...
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### What is this probability distribution?

Thank you in advance for any suggestions or feedback. I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$. I am interested in finding the Wasserstein (...
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### How does the sample variance change if you take subsets of $n$ observations from the original data?

Suppose $X$ is a continuous random variable for the weight of silver pennies. We then measure 338 pennies (in grams), leaving us with 338 observations. The observed mean weight is 15.722 grams and the ...
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### Variance identities given $E(x)<\infty$

ok i got two identities i want to prove (true or false) $Var\left [ \left ( X-E(X) \right )\frac{1}{E(X)} \right ]=\frac{E(X^2)-E(X)^2}{E(X)^2}$ prove since $Var(aX)=a^2Var(X)$ , $Var(a+X)=Var(X)$...
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### Mixed vs Fixed effects model

I'm currently weighing up the most appropriate of two models, one including individual as a random effect and the other discarding it: - ...
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### Upper bound for absolute third central moment

Suppose $X\in \mathbb{R}$ is a random variable with expected value $\mathbb{E}X = \mu$. I ran across a proof which uses the inequality $$\mathbb{E}[|X - \mu|^3] \leq 2^3 \mathbb{E}|X|^3.$$ Can ...
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### What are the random variables that constitute i.i.d. samples?

Consider the following passage from Wooldridge (2010): "For much of this book we adopt a random sampling assumption. More precisely, we assume that (1) a population model has been specified and (...
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### How to ascertain positive correlation between random variables?

Suppose we have random variables~$X,Y,Z$ that are related by the following: $$X = Y + f(Z)$$ for some function $f$. Under what conditions on the random variables and $f$ ...
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### When I repeat an experiment 10 times, do I have 10 different random variables all of which are from that sample space?

I toss 2 dice to get their sum; now, this means I have a sample space from 2 to 12. Now, when I repeat this experiment 10 times, do I have 10 different random variables all of which are from that ...
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### Derivation of Pearson's Product Moment Correlation Coefficient's (PPMCC) distribution from bivariate normal variables?

I'm interested in reading through the derivation of the probability density function that PPMCC follows when it's input is a bivariate normal variables. Mathworld gives the following equalities: P(r)...
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### Is this problem calculable only due to the parameter choices?

I am looking at a problem form Hogg, Tannis & Zimmerman (Ed. 10), and I am curious if the given problem is calculable (for an upper-level undergrad math/stats course) because of the choice of the ...
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### Show $X_n \to 0$ in probability

I am asked to show : Let $X$ be a real-valued random variable on $(\Omega, F , P)$ and define $X_n(\omega) = nX(\omega)$ if $n<X(\omega)\le n+1$ and $0$ if else. Prove that $X_n \to 0$ in ...