Questions tagged [random-variables]

Questions about maps from a probability space to a measure space which are measurable.

A random variable $X: \Omega \to E$ is a measurable function from a set of possible outcomes $\Omega$ to a measurable space $E$. The technical axiomatic definition requires $\Omega$ to be a sample space of a probability triple. Usually $X$ is real-valued.

The probability that $X$ takes on a value in a measurable set $S \subseteq E$ is written as :

$$P(X \in S) = P(\{ \omega \in \Omega|X(\omega) \in S\})$$

where $P$ is the probability measure equipped with $\Omega$.

10873 questions
20
votes
2 answers

Summing (0,1) uniform random variables up to 1

Possible Duplicate: choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries? So I'm reading a book about simulation, and in one of the chapters about random numbers…
Haile
  • 417
  • 1
  • 4
  • 10
20
votes
6 answers

What does the value of a probability density function (PDF) at some x indicate?

I understand that the probability mass function of a discrete random-variable X is $y=g(x)$. This means $P(X=x_0) = g(x_0)$. Now, a probability density function of of a continuous random variable X is $y=f(x)$. Wikipedia defines this function $y$ to…
jII
  • 2,715
  • 2
  • 20
  • 33
19
votes
2 answers

Comparing two exponential random variables

Let $A$ and $B$ be independent random variables drawn from the exponential distribution with parameters $\lambda_A<\lambda_B$. What is the probability that $A
python55
  • 2,085
  • 1
  • 8
  • 24
18
votes
2 answers

Why "One cannot construct more than countably many independent random variables"?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a nontrivial way, neither of them concentrated on a …
kerzol
  • 550
  • 2
  • 13
18
votes
3 answers

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?
wtsang02
  • 345
  • 1
  • 2
  • 10
17
votes
1 answer

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s \in [0,1])$ be a fractional Brownian motion (here…
17
votes
4 answers

Question on the 'Hat check' problem

The famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to find the expected number of men who get their right…
17
votes
3 answers

Existence of iid random variables

In probability theory we often used the existence of a sequence $(X_n)_n$ of independent and identically distributed random variables. This was already discussed here. One of the answers says: As Ahriman has pointed out, if you are given a random…
saz
  • 112,783
  • 10
  • 122
  • 213
17
votes
4 answers

What does "identically distributed" mean?

When two distributions have the same variance and shape, do we call them identically distributed, regardless of their mean (as this is usually a location parameter)?
16
votes
1 answer

Why is the function $\Omega\rightarrow\mathbb{R}$ called a random variable?

I do not understand the relation of a normal variable "x", which is to me just a placeholder for an element of a set, and a random variable, which is a mapping from the set of all possible events to $\mathbb{R}$. To make the question more concrete,…
user13247
16
votes
3 answers

Difference of two binomial random variables

Could anyone guide me to a document where they derive the distribution of the difference between two binomial random variables. So $X \sim \mathrm{Bin}(n_1, p_1) $ and $Y \sim \mathrm{Bin}(n_2, p_2) $, what is the distribution of $|X-Y|$. thank…
kolonel
  • 1,082
  • 1
  • 9
  • 22
16
votes
1 answer

Uniform distribution on a simplex via i.i.d. random variables

For which $N \in \mathbb{N}$ is there a probability distribution such that $\frac{1}{\sum_i X_i} (X_1, \cdots, X_{N+1})$ is uniformly distributed over the $N$-simplex? (Where $X_1, \cdots, X_{N+1}$ are accordingly distributed iid random variables.)
charles.y.zheng
  • 1,183
  • 1
  • 9
  • 14
16
votes
5 answers

How is the derivative of the CDF of a random variable $X$ its PDF?

For a continuos random variable $X$, if we have its p.d.f. $f(x)$, then the cumulative densitity function (c.d.f.) of $X$, $F(x)$, is $$F(x) = \int_{-\infty}^{x}f(t)dt$$ We also have $$F'(x) = \frac{d}{dx}F(x)= f(x)$$ Why does the last step in the…
16
votes
2 answers

Is there a sense in which the Chi-squared distribution is an inner product?

I have been self-studying statistics recently, and the apparent similarities between linear algebra (especially Hilbert spaces) and statistics have been popping out to me. Linear independence gets replaced with statistical independence. The mean is…