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 […]

I have a set of users that generate calls. If I assign the same probability to each user, they have identical call generation probability which can be defined as $\delta$. These callers are chosen uniformly among the set of users. At the end of the generation process, the representation of the probability density function of […]

Given 200 cards where each card has a unique number from 1 to 200. We randomly pick 30 cards (the order we pick them matters). What is the probability the unique numbers of the cards we pick are in ascending order?

A box contain $A$ white and $B$ black balls and $C$ balls are drawn, then the expected value of the number of white balls drawn is ? The answer is $\large \frac{ca}{a+b}$. How to approach this one?

Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number value while the person who has a larger number will halve their number value. Disregard any draws. For example, if […]

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