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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23 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
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2answers
57 views

Expected value of the sum of the two largest values from a Uniform parent

Is the expected value of the sum of two greatest values in an uniform distribution in [0,1] of n random variables (x1,x2,x3,x4,...,xn) equal to E(max(x^n))+E(max(x^(n-1)))?
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3answers
28 views

Find PDF of the random variable Y. Please help!

Let $X_1$ and $X_2$ be independent and identically distributed Uniform $(0,1)$ random variables. Let $Y = \max(X_1, X_2)$. Find the PDF of the random variable $Y$. I am having a hard time progressing ...
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0answers
10 views

Even dirstibution of a small set of random choices into a small set of buckets

Is there a way to evenly distribute randomly selected small set of items from a relatively larger set into to a small number of buckets using a hash function? For ex: Randomly select 20 numbers from ...
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3answers
90 views

$P\{B^{2}-4AC\geq 0\}$ where $A,B,C \sim U(0,1)$?

The actual problem is to find the probability that $Ax^{2}+Bx+C=0$ has real roots. This boils down to whether or not the discriminant $B^{2}-4AC$ is non-negative. Thus, we seek $P\{B^{2}-4AC\geq 0\}$. ...
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26 views

Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
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2answers
43 views

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?
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30 views

Uniform Sampling over Convex Polytope (not full-dimensional)

I want to simulate a uniform distribution on a convex polytope that is not full-dimensional for optimization purposes (to generate random points on the set I want to minimize over). The polytope is ...
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35 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
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1answer
577 views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, ...
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1answer
29 views

Approximation of uniform distribution.

There are leaving from the station arriving every 10 minutes. A person has to wait from 0 to 10 minutes at the station, this is uniformly distributed. Now if the person uses the station 100 times a ...
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1answer
25 views

Shuffeled coin tossing

Had a small question: Let's consider the probability space $(\Omega, \mathfrak{F})=([0,1], \mathfrak{B})$ with Lebesgue measure $\mathbb{P}$, $\mathfrak{B}$ is Borel sigma algebra. Lets expand a ...
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0answers
31 views

probability: continuous uniform distribution mean by symmetry

I am trying to show that the mean of the uniform continuous distribution is $(b+a)/2$ by symmetry. The direct method is fairly simple but, for some reason, I cant get this one. \begin{align} E[X] ...
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1answer
406 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 ...
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1answer
50 views

probability: nonlinear best predictor $\hat{Y} = g(X)$

Consider $X\sim\mathcal{U}(-1,1)$ and $Y = X^2$. The nonlinear predictor is defined as $$ \hat{Y} = g(X) = E_{Y|X}[Y|x_i] $$ Now $E_{Y|X}[Y|x_i] = \int_{-\infty}^{\infty}y\frac{f_{X, Y}(x, ...
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1answer
24 views

probability: best linear predictor $\hat{Y} = aX + b$

Let $X\sim\mathcal{U}(-1, 1)$ and $Y = X^2$. Since the best linear predictor is defined as $$ \hat{Y} = E_Y[Y] + \frac{\text{cov}(X, Y)}{\text{var}(X)}(x - E_X[X]) $$ Can I simple just write it as ...
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1answer
29 views

Determine probability of fewer than a certain number of events

Could anyone help with the following problem? My guts is telling me that the answer to part (a) is a normal distribution. Mainly, because I can't see where a uniform distribution would fit in this ...
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0answers
8 views

Using the rejection method to generate values

I'm trying to solve the following problem on rejection sampling: I think I have a good idea about what rectangle I should be using. In my mind, it would be a rectangle just large enough to encompass ...
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13 views

Calculate the marginal p.d.f of Y, f2(y), when X~U(0,2) and the conditional distribution of Y, given X = x is U(0,x^2)

Let X have a uniform distribution U(0,2), and let the conditional distribution of Y, given that X = x, be U(0,x^2). a) Determine f(x,y), the joint p.d.f. of X and Y: f1(x)=1 0<=x<=2 ...
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1answer
23 views

continuous probability: signal to noise ratio $\mu^2/\sigma^2$

$\DeclareMathOperator{\var}{var}\DeclareMathOperator{\cov}{cov}$ The signal-to-noise ratio (SNR) of a random variable quantifies the accuracy of a measurement of a physical quantity. It is defined ...
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9 views

UMP test for two different distributions

My question: UMP(Uniformly Most Powerful) test for $H_0: X\sim \mathcal{U}(0,1)$ vs $H_1: X\sim \mbox{Exp}(1)$. My attempt : By Neyman Pearson lemma, the best critical region is $Y \ge c$ Where $Y$ ...
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1answer
11 views

Find percentage of conforming items for normal and uniform distributions

I'm given the following problem: Could anyone give some insight into how to solve this? I understand that for the normal distribution, I will likely have to look the percentage up in a table. ...
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2answers
51 views

assume that X and Y are independent with X ~ UNIF(-1,1) and Y~UNIF(0,1).

I am trying to find the probability that the roots of the equation h(t)=0 are real, where h(t)=t^2+2Xt+Y of the given data. I know that I need to look at Uniform continuous distributions but I am ...
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2answers
30 views

Uniform Probability Distribution 1

A manager of a department store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 2 to 4 minutes. If it takes the elevator 30 ...
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1answer
33 views

How can I prove that Xn converges to 0 in probability?

Let $X_n\sim U[-1/n,1/n]$. Since for convergence in probability for every $\epsilon>0$, $$ \lim_{n\to\infty} P(|X_n - X|\ge \epsilon) = 0 $$ Hence, $P(|X_n-0|\ge ...
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1answer
26 views

Expectation of continuous uniform distribution

I'm having a problem with a basic probability problem. There is a stick which is 4 units in length, we break it in two pieces and the breaking point is randomly distributed. After this we form a ...
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1answer
256 views

Conditional uniform distribution

I had this question in a quiz, and now that I am reviewing it, I am not sure if why my TA gave me the marks because I am pretty sure I am wrong. Let the r.v. $Y$ follow uniform distribution $U(1,2)$ ...
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1answer
252 views

Weakly bounded iff uniformly bounded in $E'$?

I have a problem: Suppose that $E$ be a normed space over $\mathbb{R}$ and $E= \{f: [0,1] \to \mathbb{R}\ \text{is continuous and such that}\ f|_{[0, \delta]}=0, \text{with}\ \delta=\delta(f)>0 ...
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2answers
61 views

A random variable $X$ uniformly distributed over the interval $[0, 2\pi]$

A random variable $X$ distributed over the interval $[0, 2\pi]$ a) the pdf of $X$ b) the cdf of $X$ c) $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ d) $P(-\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ ...
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1answer
20 views

Conditional probability explained

Sorry for the dumb question, but it seems that I'm missing something pretty straightforward Abstract Suppose you are throwing one cube of dice, and you have thrown value "6" ten times in a row, ...
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1answer
58 views

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]?

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]? $E[e^{\frac{2X}{3}} - 3] = \int_0^2 \! e^{\frac{2X}{3}} - 3 \, \mathrm{d}x$ $= \frac{3}{2}(e^{\frac{4}{3}} - 5) = -1.8095$ I am integrating over the ...
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4answers
55 views

Expectation of product of cosine and sine

$\theta\sim U(-\pi,\pi)$. When $\theta$ follows uniform distribution, what is the expected value of the producot of cosine and sine, i.e. $$E[\sin\theta \cos\theta] = \ ?$$
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23 views

Uniform Distribution Min & Max unbiased estimators?

I have $n$ samples ($iid$) from a uniform distribution over $[a,b]$ with unknown $a$ & $b$, let call them $x_i$. Let $Min\{x_i\}=\hat{a}$ and $Max\{x_i\}=\hat{b}$. What are unbiased estimators ...
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1answer
35 views

Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$

Let $m$ be an arbitrary value in $Z_n$, where n is RSA modulo (n=p.q, where p and q are large primes). Then have: $r_2=r_1 . m$, where $r_1$ is a value chosen uniformly at random : $r_1\in Z^*_n$. ...
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0answers
13 views

orderstatistics of uniform distributions on different ranges

During a simulation I discovered an interesting phenomenon: Given you have 3 agents. 2 are uniformly distributed between [0,1] and one between [0,2]. The question is how often do the smaller agents ...
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1answer
18 views

Distributing years in a Leap Year system.

First let's take our Leap Year system: a Leap Year is a year of 366 days, as opposed to normal years of 365 days. It occurs every 4 years, except if the year is divisible by 100, and not divisible by ...
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16 views

Question about the uniform distribution and random variable operation

If I have a random variable with uniform distribution ranging from 0 to 1, as follows: $ X \sim U\left ( 0 ;1 \right ) $ And if I subtract that random variable $ X $ from 1, will the resulting random ...
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2answers
32 views

Question about probability of a random variable with uniform distribution

Why is it that when a random variable has a uniform distribution the following statement it's true? $ \Pr \left ( X_{1} \leqslant c \right ) = c $ The question arised when I was doing this ...
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1answer
57 views

Question about a sequence of iid random variables and the Uniform distribution

I will first enuntiate the question and then explain what I'm not understanding. Suppose $ X_1, X_2,\ldots, X_n $ iid with common distribution $ U(0,\theta)$. Define $M$ as follows: $ M : =\max\{ ...
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40 views

Probability that one uniform distribution is less than another

I am trying to pick the variable r which maximizes an expected return. So I need to calculate the probability that $rn < x$ where n and x are both random variables with uniform distributions ...
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1answer
36 views

Is it possible to generate a uniformly distributed random 128-bit number from multiple uniformly distributed random numbers of size <= 32 bits?

If I have a uniformly distributed random number generator of up to 32 bits in length, can I generate a uniformly distributed 128 bit number by rolling my 32-bit random number generator multiple times ...
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1answer
31 views

Average distance between consecutive uniformly distributed values

Say I have a list of values $(X_1, X_2, ..., X_n)$ and $X_i$ is a uniform random variable between 0 and 1. Let $(Y_1,Y_2,...,Y_n)$ be the ordered list of those values. How do I find the expected ...
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0answers
25 views

Location, scale and location scale family

Let $X_1,...,X_n$ be an i.i.d. random sample from a continuous uniform distribution over [0,$\theta$], where $\theta>0$ is a unknown parameter. Does these random variables belong to a family of ...
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1answer
52 views

Roll a die. Are the chances of getting two same consecutive numbers the same as getting any specific random sequence?

Let's imagine a die roll. You roll the die n times. Example: I have a $6$ sided die. Assuming the distribution of the die is perfect, the chances of getting any single number are $1/6$. The ...
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1answer
85 views

About strong law of large numbers

I came across a problem: Let $X_1,X_2,...$ be independent random variables with uniform(0,1) distribution from probability space $(\Omega,\cal F,P)$. Prove that: ...
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2answers
101 views

Uniform distribution with random support

I have a uniform distribution $ X \sim U(A,B) $ where the limits themselves are random: $A \sim N(\mu_A,\sigma_A^2)$ and $B \sim N(\mu_B,\sigma_B^2)$. Hence the support of $X$ is random. $A$, $B$ are ...
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0answers
33 views

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,...,n$ Determine the limit ...
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0answers
17 views

Standard Uniform Distibution with Random Variable

Could someone help explain how to solve the following problem: From my understanding, this problem states that we have a function, Uniform(0, 1), that will generate a random value from 0 to 1 with ...
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1answer
228 views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in\{-\alpha,\alpha\}$, $Y\in\{-\alpha,\alpha\}$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
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62 views

To make a polynomial with coefficients in a finite field uniform at random

We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$. Let $P_1$ be a polynomial such that $P_1 \in R[x]$. The aim is to compute $P_2=P_1 . r$, where ...