Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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).

As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.

Reference: Wikipedia

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Generate a random variable with a defined correlation to an existing variable(s)

For a simulation study I have to generate random variables that show a predefined (population) correlation to an existing variable $Y$. I looked into the R packages copula and CDVine which can produce random multivariate distributions with a given…
Felix S
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Convergence in probability vs. almost sure convergence

I've never really grokked the difference between these two measures of convergence. (Or, in fact, any of the different types of convergence, but I mention these two in particular because of the Weak and Strong Laws of Large Numbers.) Sure, I can…
raegtin
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What is meant by a "random variable"?

What do they mean when they say "random variable"?
Baltimark
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What are i.i.d. random variables?

How would you go about explaining i.i.d (independent and identically distributed) to non-technical people?
user333
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How is the minimum of a set of IID random variables distributed?

If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
Simon Nickerson
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Variance of product of multiple independent random variables

We know the answer for two independent variables: $$ {\rm Var}(XY) = E(X^2Y^2) − (E(XY))^2={\rm Var}(X){\rm Var}(Y)+{\rm Var}(X)(E(Y))^2+{\rm Var}(Y)(E(X))^2$$ However, if we take the product of more than two variables, ${\rm Var}(X_1X_2 \cdots…
damla
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Why does the correlation coefficient between X and X-Y random variables tend to be 0.7

Taken from Practical Statistics for Medical Research where Douglas Altman writes in page 285: ...for any two quantities X and Y, X will be correlated with X-Y. Indeed, even if X and Y are samples of random numbers we would expect the…
nostock
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Intuitive explanation for density of transformed variable?

Suppose $X$ is a random variable with pdf $f_X(x)$. Then the random variable $Y=X^2$ has the pdf $$f_Y(y)=\begin{cases}\frac{1}{2\sqrt{y}}\left(f_X(\sqrt{y})+f_X(-\sqrt{y})\right) & y \ge 0 \\ 0 & y \lt 0\end{cases}$$ I understand the calculus…
lowndrul
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How can I efficiently model the sum of Bernoulli random variables?

I am modeling a random variable ($Y$) which is the sum of some ~15-40k independent Bernoulli random variables ($X_i$), each with a different success probability ($p_i$). Formally, $Y=\sum X_i$ where $\Pr(X_i=1)=p_i$ and $\Pr(X_i=0)=1-p_i$. I am…
David B
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Intuitive explanation of convergence in distribution and convergence in probability

What is the intuitive difference between a random variable converging in probability versus a random variable converging in distribution? I've read numerous definitions and mathematical equations, but that does not really help. (Please keep in mind,…
nicefella
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What is it meant with the $\sigma$-algebra generated by a random variable?

Often, in the course of my (self-)study of statistics, I've met the terminology "$\sigma$-algebra generated by a random variable". I don't understand the definition on Wikipedia, but most importantly I don't get the intuition behind it. Why/when do…
DeltaIV
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Brain-teaser: What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?

This is an interview question for a quantitative analyst position, reported here. Suppose we are drawing from a uniform $[0,1]$ distribution and the draws are iid, what is the expected length of a monotonically increasing distribution? I.e., we…
Amazonian
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Simple examples of uncorrelated but not independent $X$ and $Y$

Any hard-working student is a counterexample to "all students are lazy". What are some simple counterexamples to "if random variables $X$ and $Y$ are uncorrelated then they are independent"?
Clare Brown
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Variance of a function of one random variable

Lets say we have random variable $X$ with known variance and mean. The question is: what is the variance of $f(X)$ for some given function f. The only general method that I'm aware of is the delta method, but it gives only aproximation. Now I'm…
Tomek Tarczynski
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Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence between the sample mean and the sample variance of…
JohnK
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