Questions tagged [uniform-distribution]

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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Normal Distribution and Dice game

Suppose I use the following rules to generate random numbers. I roll a fair $D_6$ twice. The first roll I call $\mu$ and the second $\sigma^2$. I then Generate a normal random variable with mean $\mu$ and variance $\sigma^2$. I call this $X$ In more…
user524813
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Probability depending on two uniformly distributed variables

Apologies if this is a duplicate, the problem is very hard to word in a search The bivariate distribution of random variables X and Y is uniform over the triangle with vertices (1,0), (1,1), (0,1). A pair of values $x,y$ is chosen at random from…
janes
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A problem with uniform distribution?

In a game Alis and Daniel shoot arrows on a circular target with a radius of R. Alis throws arrows such that their distance from the center have Continuous uniform distribution (0,R) - o,R are the parameters for uniform distribution. While Daniel…
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Inverse transformation sampling: Logistic CDF

We have a generator $X$ that selects numbers from a uniform distribution on $(0,1)$ denoted $\text{Unif}(0,1)$. We have to show how it can be used with the function $x \mapsto \log\left(\frac{x}{1-x}\right)$ to generate a random number from a…
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Generate a uniform distribution of orientations

I am rendering images of a 3D object and am trying to generate a uniform distribution of orientations for this object. I am unsure of how to do this, and considerations that are necessary to take into account. Is a uniform distribution of…
Acoop
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How do I find the transformation between the following random variables?

Let $U$ $\sim$ $U(0,1)$ . The Probability Integral Transform theorem states that for any continuous random variable $Y$ with cumulative distribution function $F(y)$ and inverse cumulative distribution function $F^{-1}$: $F(Y)$ is a $U(0,1)$ random…
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Find the variance of $U[a,b]$ from $U[0,1]$

I know the variance of uniform distribution $U[a,b]$ is $(b-a)^2/12$, but I'm having trouble to prove it from $U[0,1]$. i.e. let $X = a+(b-a)U[0,1], Var(X) = \mathbb{E}(X^2)+\mathbb{E}(X)^2 = \mathbb{E}(a^2+(b-a)^2U^2+2a(b-a)U)-(a+b)^2/4 =…
smaillis
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Modulus of a random variable which follows a continuous uniform distribution, follows which distribution?

Let $X \sim U(\theta,0),$ $\theta<0$ (continuous uniform distribution) I want a transformation on $X$ so that it follows $U(0,1)$ distribution I did $|X|/\theta\sim U(0,1).$ Am I right?
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How to show that something has the same probability distribution function as something else?

I understand the logic behind what I'm about to ask, but I'm not sure exactly how to write it mathematically. I have a random variable $X$ with probability density function: $$ f_X(x)= \begin{cases} x^{-2} & x\ge 1\\ 0 & x<1 \end{cases} $$ and…
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Uniform distribution over area of a dartboard vs over distance from center

A bad player hits a dartboard represented by the unit circle with uniform probability over its area, and a good player has uniform distance distribution over [0,1]. But what's the difference between the two, if every point in the circle is some…
Melanie
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Joint probability of uniform random variables and absolute difference

Let $X,Y$ be independent standard uniform random variables. How do we compute the joint density $$P(a\leq X\leq 1-a, a \leq Y \leq 1-a, |X-Y|\geq a) $$ for some positive value $a$. I can split the absolute value into two separate conditions $X-Y\geq…
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Probability of meeting using integral

Here is a famous question- Two people agree to meet sometime between 9am and 10am. Each picks a time uniformly and waits for 15 minutes. What is the probability that they meet? I know that this can be easily tackled by drawing rectangles. The…
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Let $X\sim \text{Uniform}(0, \theta)$. What is the distribution of $Y= \theta - X$?

So cdf of $X$ is $F(X\leq x) = \frac{x}{\theta}$. Let $X = \theta - Y$, subbing it into the cdf we now get: $$F(\theta - Y \leq x) = \frac{x}{\theta}$$ $$F(-Y \leq x - \theta) = \frac{x}{\theta}$$ $$F(Y\geq \theta - x) = \frac{x}{\theta}$$ $$F(Y\leq…
doctopus
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Marginal Density and Conditional Density Problem

Suppose (X, Y) is uniformly distributed over the region defined by $0 \leq y \leq \sqrt{1-x^2}$ and $-1 \leq x \leq 1$. a. Find the marginal densities of X and Y b. Find the conditional densities of $f_{X|Y = y}(x) and f_{Y|X = x}(y)$. c. Are X and…
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Transformation of Uniform Distribution using $Y = X^2$

I know this has already been answered somewhere but I can't seem to find where I am going wrong. $X$ has a uniform distribution on $[-1,1]$ Let $Y = X^2$ P.d.f of X: $f_X(x) = \frac{1}{2}$ for $-1 \le x \le 1$ Then using $f_Y(y) = f_X(g^{-1}( y))$…
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