Probability

So I have a sequence of n_t frames and each frame has n dots with coordinates (x_n,y_n). Most of the points are drawn at random, but there are two points are actually not random and move with constant...

Suppose $f:A\to\mathbb{R}$ and $A$ is countably infinite subset of the real numbers. We can enumerate $A$ as $\left\{a_n\right\}_{n=1}^{\infty}$ and set $t$ as a natural number but the average \lim\li...

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Three red balls and three green balls are randomly placed on a circle. Find the expectation of the number of red balls such that both their neighbours are green.check pic please

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Let $\vec{X}\sim N_d(\vec{\mu},\Sigma)$.  If $\vec{\mu}=\mu \vec{1}$ and $\Sigma=\sigma^2 I_d$, find the distribution of $\overline{X}_d$ conditioned to the event \{d\max\{X_1,\dots,X_d\}\in B\}, wher...

A researcher conducted an experiment to evaluate if the proportion of supporters for the most corrupted party in the country is equal to 40% (null hypothesis). The alternative hypothesis is that the p...

I'm working on a board game and am trying to check the balance of a mechanic. Specifically, I'd like to calculate the profit a player is likely to make by giving out a loan. Each loan has four values:...

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. The width of a rectangle X is a random variable uniformly distributed on the interval (1,2) , whereas its height Y is uniformly distributed on the interval between 1/2 and X/2 . Find the expected ar...

Just check if the problems are correct, and provide a the correct answer if it isnt. Problems are on the screenshot

In a casino, two differently loaded but identically looking dice are thrown in repeated runs.The frequencies of numbers (number of times each number has been observed) observed in 40 rounds of play ar...

A company manufactures metal poles. Suppose the length of a pole is a random variable X, with mean $\mu _X$ and probability density function $f_X(x)$ . Poles are cut to obtain an exact length 𝐿. If th...

Show that for any $\phi \in C_0(\mathbb{R^n})$ (a continuous function on $\mathbb{R^n}$ which tends to 0 at $\infty$), if we set $\phi_\epsilon(x) = \mathbb{E}\phi(x + \epsilon Z)$ for any random vari...

Find the maximum likelihood estimator

Find the maximum likelihood estimate

Find the conditional mean and variance

Let $\{X_n\}$ be a sequence of random variables such that $X_n\overset{P}{\to}X$ for some random variable $X$. Suppose that for some $p\geq1$,  \lim\limits_{a\to\infty}\mathbb{E}\left[ |X_n|^p\mathbb{...

The Problem-There are 5850 total tickets. - 8 tickets will be randomly chosen at the same time.- Once a ticket is chosen, the same ticket number cannot get chosen again. - I have 290 tickets or 3.57%...

Let $\{X_n\}$ be a sequence of random variables and $S_n:=\sum_{j=1}^n X_j$. Proof that if $X_n\overset{\mathcal{L}_p}{\to}0$ for some $p\geq 1$, then $n^{-1}S_n\overset{\mathcal{L}_p}{\to} 0$ but the...

Five married couples are seated around a table at random. Let $X$ be the number of wives who sit next to their husbands. What is the expectation of $X$ $(E[X])$?