2
votes
1answer
34 views

$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$ independent and normal

I am looking for a nice proof of the following statement: If $X,Y\sim U(0,1)$ are two independent uniformly distributed random variables, then $$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), \quad ...
1
vote
1answer
55 views

Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...
0
votes
0answers
94 views

Geometric Mean of Uniform random variables convergence

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
0
votes
0answers
89 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
0
votes
1answer
330 views

Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: ...
1
vote
1answer
66 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
0
votes
0answers
90 views

Summing many non-standard i.i.d. uniform random variables

all! I have looked up a fair bit on this question and learned much about the problem. But haven't been able to get any crisp answers. Sorry, if I'm missing something obvious. I know one can use the ...
1
vote
2answers
55 views

$X$ is half normal and $S ∼ U{(−1, +1)}$. How $Z = SX ∼ N(0, 1)$?

If we chop a standard normal distribution in half and use only the positive side (scaled up by a factor of $2$ to maintain a proper density), then we get the so-called ‘half normal’ density: ...
2
votes
1answer
340 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
0
votes
0answers
57 views

Is transfert theorem the best choice in this kind of exercise?

I am studying Probability theory and came to this exercise : Let $U,V$ be independent uniform random variables over $[0,1]$. Show that $X:=\cos(2\pi V)\sqrt{-2\ln U}$ and $Y:=\sin(2\pi V)\sqrt{-2\ln ...
0
votes
1answer
596 views

equivalence between uniform and normal distribution

The principle of insufficient reason says that all outcomes are equiprobable when we have no knowledge to guess otherwise. I understand this and that this corresponds to uniform distribution. However, ...
1
vote
1answer
328 views

Normal Random Variable - uniform distribution

So here's the question I'm trying to solve: A stock price movement model supposes that if the current stock price is s, then, after one period, the stock price will be $us$ with probability ...
3
votes
1answer
146 views

Estimating number drawn from one distribution based on sum of that number and number drawn from another distribution

I have been working on this for several days and have been unable to come up with an answer. The problem is very simple to state, but it seems difficult to solve. A computer draws a number $x$ at ...