# Tagged Questions

For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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### Pdf for distance between two uniform random points in a circle

This is my first post in the group and I would be very thankful for any help. I am trying to develop a probability distribution for a performance analysis in my thesis. I am trying to look in to ...
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### Show that $\mathbb{P}(|\frac{1}{n}\sum_{i=1}^nX_i-\frac{1}{2}|>\frac{1}{2})\leq\frac{1}{3n}$

Suppose $X_1, X_2, ..., X_n$ are independent uniformly distributed random variables on [0,1]. Show that $\mathbb{P}(|\frac{1}{n}\sum_{i=1}^nX_i-\frac{1}{2}|>\frac{1}{2})\leq\frac{1}{3n}$ I've ...
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### How do I evenly distribute some time? [on hold]

Here's my problem. I have a device (ozone generator) which produces 10g of ozone per hour, or 2.77778 milligrams per second (I think). I need to be able to control the production per hour by pulsing ...
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### Probability of even sum of $n$ integers with uniform distribution from $\{1,2,\dots, 2n\}$.

Choosing with Uniform distribution $n$ numbers from $\{1,\dots,2n\}$ with returns and the order is important. What is the probability that the sum of these number will be even? Thanks.
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### Probablity of normal distribution when x is a function

Assume a uniform distribution random variable X~U(0,1). And $\Phi$ is the symbol of the standard normal distribution. Assume $Y=\Phi^{-1}(X)$. The question is, $\mathbb{P}(Y \le 0)=?$. The Solution is ...
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### Uniform Distribution Problem on $X, Y, Z$

Problem: Let $X \sim \text{Uniform}(0,1)$. Let $0 < a < b < 1$. Let $$Y = \begin{cases} 1 & 0 < X < b \\ 0 & \text{otherwise} \end{cases}$$ ...
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### Probability that a set of uniformly distributed random variables is 'greater' than another such set.

Suppose we generate several uniformly distributed random variables (between 0 and 1), and arrange them in descending order to form a set [A1, B1, C1...]. We then do the same process to form a second ...
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### Determine $\lim_{n\to\infty} \mathbb{P}(\sum_{i=1}^n X_i \leq \frac{n}{2})$

Suppose $X_1, X_2, ... X_n$ are independent and uniformly distributed (on $[0,1]$) random variables. Determine $\lim_{n\to\infty} \mathbb{P}(\sum_{i=1}^n X_i \leq \frac{n}{2})$ My thoughts were the ...
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### Distribution function of uniform distribution

I don't know why the distribution of this question is that when x is in between 0 and theta. In solution.. is that right? I searched the distribution of uniform distribution. But it is alike with that ...
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### Integration Question Arising from Attempting to Compute a Marginal Distribution

I have a question about the last step of a proof on page 37 of All of Statistics. The entire proof is here for sake of completeness, but I don't think grokking it in its entirety is necessary for ...
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### Change of Uniform Continuous Variable

Let $X$ be a $U(-1, 1)$ random variable, we define $Y = X^4$. Calculate the correlation coefficient between both variables. Are they uncorrelated? PS. I don't know how to use MatJax equations, I'm ...
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### Meeting probability of two bankers: uniform distribution puzzle

Two bankers each arrive at the station at some random time between 5PM and 6PM (arrival time for each of them is uniformly distributed). They stay exactly five minutes and then leave. What is the ...
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### Uniform random variables, length of smallest interval around given point

The following claim is stated without proof/reference in something I am reading: Let $X_1,\ldots,X_n$ be i.i.d. uniform on $[0,1]$, and let $c \in (0,1)$ be fixed. If $Z = \min\{X_i : X_i > c\}$...
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### Choose $x$ objects without replacement from a bag with $n$ object.

General problem: Suppose there is a bag containing $n$ items with $m$ unique values $(m \leq n)$. The distribution of values across all the items is uniform. How many unique values I most probably ...
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### Points in hemisphere over plane defined by a normal vector

I have the following formulas to sample points uniformly on a unit sphere in 3D space: $x = \sqrt{1-u^2} sin\phi$ $y = \sqrt{1-u^2} cos\phi$ $z = u$ where $u \in [-1,1]$ and $\phi \in [0,2\pi]$. ...
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### Probability of rectangles area being less than 0.5 w/ total length of sides = 2

Question: A random point splits the interval [0,2] in two parts. Those two parts make up a rectagle. Calculate the probability of that rectangle having an area less than 0.5. So, this is as far as I'...
This is a qual problem。 Let $n$ points be iid uniformly distributed on the unit circle. Let $\Delta_n$ be the smallest distance between any two of these points. Show that $n^{\theta}\Delta_n\... 0answers 25 views ### Density probability I ask myself a question about of density next : p(xi)=1/(pi*(x²+1)) The law marginal is easy to identify of X and Y: ... 0answers 29 views ### Find Expected Value, Variance, and Limit of Uniform Distribution Let$X_1, X_2, \ldots, X_n$be a sequences of independent random variables.$X_i \sim U(0, 2A)$. Compute$E(X_i)$and the$Var(X_i)$. Also compute the$lim_{n\to\infty} P(X_1, X_2, \ldots , X_n > ...
Suppose that $𝑋_1$ and $𝑋_2$ are independent and follow a uniform distribution over $[0, 1]$. Let $𝑌_1 = 𝑋_1 + 𝑋_2$, and $𝑌_2 = 𝑋_2 − 𝑋_1$. a) Find the joint pdf $𝑓_{𝑌_1,𝑌_2} (𝑦_1, 𝑦_2)$ ...