I am missing something that might be trivial in deriving the mean of the geometric distribution function by using expected value identity $$ \sum_x x \theta (1-\theta)^{x-1}. $$

Suppose that $X_{\sigma} \sim \mathcal{N}(\mu,\sigma^{2})$. I am interested in whether $f(\sigma)=\mathbb{E} (X_{\sigma}^2 1_{\{X_{\sigma}>0\}})$ is monotonic in $\sigma$ for all $\mu$. I ran a Monte Carlo and it does appear monotonic but I can’t rigourous prove it.

Let $(\pi_1, \pi_2, \cdots)$ be an infinite sequence of real numbers such that $\forall i\; \pi_i > 0$ and $\sum_i \pi_i = 1$. This can be thought of as a probability over natural numbers. Let $(z_1, z_2, \ldots)$ be a sequence of independently and identically distributed Bernoulli random variables such that $P(z_i = 1) = […]

I have the PDF $f(x)=\frac{3}{4}(1-x^2)\mathbf 1_{-1<x<1}$ and accordingly the CDF $$F(x)=\begin{cases}0, &\phantom{-}x\le -1\\\frac{3}{4}x-\frac{1}{4}x^3+\frac12, & -1<x<1 \\1, & \phantom{-}1\le x\end{cases}$$ Since the formula for $\mathbb{E}[X]$ is the integral of $S(x)=1-F(x)$ is from $0$ to $\infty$, how do I account for the fact that $x$ is only defined from $-1$ to $1$ when I need to calculate […]

Given a coin with an unknown bias and the observation of $N$ heads and $0$ tails, what is expected probability that the next flip is a head?

I was wondering how does the rules of probability apply if we have a discrete random variable $Y$ and a continuous random variable $X$. What would $P(Y=y,X=x)$ be equal to? Is joint probability defined or is it joint density function? (note: I denote probability as $P$ and density functions with $f$) Is it correct to […]

For the function $Q(x) := \mathbb{P}(Z>x)$ where $Z \sim \mathcal{N}(0,1)$ \begin{align} Q(x) = \int_{x}^\infty \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2} \right) \text{d}u, \end{align} for $x \geq 0$ the following bound is given in many communication systems textbooks: \begin{align} Q(x) \leq \frac{1}{2} \exp \left(-\frac{x^2}{2} \right). \end{align} The bound without the $\frac{1}{2}$ in front of the exponential can be proven […]

Let $X$, $Y$ be r.v. with finite second moments. Suppose $\mathbb{E}(X\mid\sigma (Y))=Y$, and $\mathbb{E}(Y\mid\sigma(X))=X$, show that $\Pr(X=Y)=1$. So what I have done is this, I first consider $\mathbb{E}((X-Y)^2)$ by conditioning on $X$ and $Y$ $\mathbb{E}((X-Y)^2\mid X)=\mathbb{E}(X^2\mid X)-2\mathbb{E}[XY\mid X]+\mathbb{E}[Y^2\mid X]=X^2-2X^2+\mathbb{E}(Y^2\mid X)=-X^2+\mathbb{E}[Y^2\mid X]$, and similarly for conditioning on $Y$, but I am not sure how to subtract […]

For any random variable $X$, there exists a $U(0,1)$ random variable $U_X$ such that $X=F_X^{-1}(U_X)$ almost surely. Proof: In the case that $F_X$ is continuous, using $U_X=F_X(X)$ would suffice. In the general case, the statement is proven by using $U_X=F_X(X^-)+V(F_X(X)-F_X(X^-))$, where $V$ is a $U(0,1)$ random variable independent of $X$ and $F_X(x^-)$ denotes the left […]

Is there a quick way to tell if a set of six-sided dice cannot be non-transitive? I’ve writing an algo and brute force is taking too long to find out. I had a look at http://math.ku.edu/~jschweig/dice.pdf but it has a precondition that numbers on a die’s face shouldn’t repeat on other faces of that die […]

Intereting Posts

Find a formula for the nth Fibonacci Number
in a mountain climbing expeditions 5 men and 7 women are to walk
Why does the semantics of first order logic require the domain to be non-empty?
Example of infinite field of characteristic $p\neq 0$
An inequality of J. Necas
Tricky positive diophantine equation
Closed form of arctanlog series
Finding out an arc's radius by arc length and endpoints
What was the notation for functions before Euler?
Proving $e^{-|x|}$ is Lipschitz
Solving a recurrence of polynomials
Logic for decomposing permutation into transpositions
Why is the Continuum Hypothesis (not) true?
Integral with Tanh: $\int_{0}^{b} \tanh(x)/x \mathrm{d} x$
Cutting a $m \times n$ rectangle into $a \times b$ smaller rectangular pieces