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 $unif(a,b)$, […]

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 […]

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 […]

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. […]

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 […]

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece

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?

Three points are chosen at random on a circle. What is the probability that they are on the same semi circle? If I have two portions $x$ and $y$, then $x+y= \pi r$…if the projected angles are $c_1$ and $c_2$. then it will imply that $c_1+c_2=\pi$…I have assumed uniform distribtuion so that $f(c_1)=\frac{\pi}{2}$…to calculate $P(c_1+c_2= […]

Is it possible to prove that the sum of two independent r.v.’s $X$ and $Y$ with convex support cannot be uniformly distributed on an interval $[a,b]$, with $a < b$? (Let us rule out the trivial case where $X$ is degenerate and $Y$ is uniform.) Note that $X$ and $Y$ need not be identically distributed. […]

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$. How do I find the PDF of $W$? How do I find the expectation of $W$ at two ways: 1. with the PDF of $W$ and without the PDF of $W$. I ask this Q before few days, but now I want show you me solution and ask if […]

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