Articles of uniform distribution

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in $\mathbb{Z}_p$?

Probability that n points on a circle are in one quadrant

Question Points $A$,$B$ and $C$ are randomly chosen from a circle, What is the probability that all the points are in one quadrant ($\frac{1}{4}$ circle)? My try Using this answer about semicircle, I tried to do the same for quadrant and I think the answer should be $\frac{3}{16}$. Is this correct? What are the other […]

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 distribution of $X_n$ as $n\rightarrow \infty$. Now I think that if I could find $f_X(x)$ and $F_X(x)$ for $X_n$ then I would have […]

Maximum of martingales

I need to show whether or not the maximum of two martingales is also a martingale. Originally, I thought yes. But supposedly the answer is no. So as a counter-example, let $U_i$ be $iid$ $unif(0,1)$, $X_0 = 1$, and $$X_n = 2^n\prod_{k=1}^n U_k.\tag{1}$$ I already know that $X_n$ is a martingale. For my second martingale, […]

Probability that, given a set of uniform random variables, the difference between the two smallest values is greater than a certain value

Let $\{X_i\}$ be $n$ iid uniform(0, 1) random variables. How do I compute the probability that the difference between the second smallest value and the smallest value is at least $c$? I’ve messed around with this numerically and have arrived at the conjecture that the answer is $(1-c)^n$, but I haven’t been able to derive […]

Derivation of Variance of Discrete Uniform Distribution over custom interval

I’m trying to prove that the variance of a discrete uniform distribution is equal to $\cfrac{(b-a+1)^2-1}{12}$. I’ve looked at other proofs, and it makes sense to me that in the case where the distribution starts at 1 and goes to n, the variance is equal to $\cfrac{(n)^2-1}{12}$. I want to find the variance of $unif(a,b)$, […]

Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn’t find the mean and variance of the order statistics of a discrete uniform sample without replacement anywhere, so I figured I’d derive them here for future reference. Draw $k$ distinct numbers uniformly from […]

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process $\left\{\tilde{B}\left(t\right): t \geq 0\right\}$ defined by $$ \tilde{B}\left(t\right) := \begin{cases} B\left(t\right), & \textrm{if } t \neq U, \\0, & \textrm{if } t = U \end{cases} $$ has the same […]

Sum of discrete and continuous random variables with uniform distribution

Could you tell me how to find the distribution of $Z = X+Y$ if $X$ is a random variable with uniform distribution on $[0,1]$ and $Y$ has uniform distribution on $\{-1,0,1\}$? $X$ and $Y$ are independent. I know how to find distributions of sums of random variables if both are discrete or both are continuous. […]

Distribution of Max(X_i) | Min(X_i), X_i are iid uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x_i) | \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b) $$ I agree that the probability density function of $\max(x_i) | \min(x_i)$ must be non-zero only in the range $[\min(x_i), b]$. I also find it reasonable that the posterior distribution is uniform, but I […]