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.

$n$ 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 […]

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? […]

Here are some examples for symmetric ones: https://mathoverflow.net/questions/55092/martingales-in-both-discrete-and-continuous-setting/55101#55101 Is there a similar list for asymmmetric random walks?

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 […]

This is a follow up to the great answer posted to https://math.stackexchange.com/a/125991/7980 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 $x^* = x^{\frac{1}{1+r}}$. […]

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 […]

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 […]

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 ?

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 […]

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