Articles of probability

Order preserve after taking expectation “piecewisely”

There are four continuous functions $\mathbb{R^+}\rightarrow \mathbb{R^+}$, $g_1(x),g_2(x),g_3(x),g_4(x)$, and they satisfy $g_1(x)g_2(x)<g_3(x)g_4(x)$ for $\forall x$, I’m wondering under what conditions about $g$ or their relationship will we have $E[g_1(x)]E[g_2(x)]<E[g_3(x)]E[g_4(x)]$ for an arbitrary probability distribution.

Likelihood Functon.

$n$ random variables or a random sample of size $n$ $\quad X_1,X_2,\ldots,X_n$ assume a particular value $\quad x_1,x_2,\ldots,x_n$ . What does it mean? The set $\quad x_1,x_2,\ldots,x_n$ constitutes only a single value? or, $\quad x_1,x_2,\ldots,x_n$ are n values , that is, $X_1$ assumes the value $x_1$, $X_2$ assumes the value $x_2$,and so on? Why Likelihood […]

A Question on Probability – Hunter and Rabbit

Suppose there are m different hunters and n different rabbits. Each hunter selects a rabbit uniformly at random independently as a target. Suppose all the hunter shoot at their chosen targets at the same time and every hunter hits his target. (i) Consider a particular Rabbit $1$, what is the probability that Rabbit $1$ survives? […]

What are some martingales for asymmetric random walks?

Here are some examples for symmetric ones: Is there a similar list for asymmmetric random walks?

Cox derivation of the laws of probability

I have read Jaynes’ Probability Theory: The Logic of Science a while ago, but mostly skimmed over parts of his derivations that I didn’t immediately understand. Now I’m trying to really understand it, but it appears he mostly skips over steps he sees as obvious or trivial. Now, I’ve been able to construct most of […]

Proving an asymptotic lower bound for the integral $\int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$

This is a follow up to the great answer posted to Let $ 0 < r < \infty, 0 < s < \infty$ , fix $x > 1$ and consider the integral $$ I_{1}(x) = \int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$$ Fix a constant $c^* = r^{\frac{1}{2r+2}} $ and let $x^* = x^{\frac{1}{1+r}}$. […]

Find $E(e^{-\Lambda}|X=1)$ where $\Lambda\sim Exp(1)$ and $P(X=x)=\frac{\lambda^xe^{-\lambda}}{x!}$.

Let $X$ have probability mass function $$P_\lambda(X=x)=\frac{\lambda^xe^{-\lambda}}{x!},\quad x=0,1,2,\ldots$$ and suppose that $\lambda$ is a realization of a random variable $\Lambda$ with probability distribution function $$f(\lambda)=\begin{cases}e^{-\lambda} &; \lambda>0 \\ 0 &; \lambda\leq 0.\end{cases}$$ What is $$E(e^{-\Lambda}|X=1)?$$ I am unsure how to interpret this question. Usually, we have either two discrete random variables $(X,Y)$, or two continuous […]

Density of Gaussian Random variable conditioned on sum

I am struggling with this simple problem. I have two Gaussian independent random variables $X \sim \mathcal{N}(0,\sigma_x^2,), Y \sim \mathcal{N}(0,\sigma_y^2,)$. I have to find the density of $X$ given that $X + Y > 0$. I know that $X, X+Y$ shall be jointly normal distributed and I also know the forms of conditional distribution of […]

Discrete random variable with infinite expectation

Consider a discrete random variable taking only positive integers as values with $$\mathbb{P}[X=n]=\frac{1}{n(n+1)}.$$ (a) Show that $\mathbb{E}[X]=\infty$. (b) Show that $\mathbb{P}[X \geq n]= \frac{1}{n}$. What does this imply for Markov’s inequality ?

Weird probability question – Grapes & olives

Does anyone know how to solve this? “A basket has 3 grapes and 2 olives. If two were taken out and random, what is the probability of picking both olives?” -My first thought is that the total number of possible outcomes would be 3 grapes(G) + 2 olives(O) = 5, but once I start writing […]