**59.578 probability questions.**

I can prove symbolically that given some random variable $X$ with a strictly increasing continuous CDF, $F$ that $Y = F(X)$ has a Uniform(0,1) distribution:
$$P(Y \leq y) = P(F(X) \leq y) = P(X \leq ...

Let's say there are random variables $A$ and $B$ being independent.
And random variable $X$.
Are there any properties to simplify $E(X | (A,B))$ : expectation of $X$ given $A$ and $B$ ?
In ...

$A$ & $B$ discrete independent variables, the probability of occurrence of each was taken as percentage $P(A)=0.6$, $P(B)=0.4$. I need to define the conditional probability tables for variable $$C ...

I came across following problem
A basketball team consists of 6 frontcourt and 4 backcourt players. If players are divided into roommates at random, what is the probability that there will be ...

It seems as though the deduction theorem can fail in natural language, if we think of a valid inference as one that preserves certainty, rather than truth. What I mean is that if we are certain of (...

A chest contains 5 envelopes that each contain one bill. One envelope
contains a $\$5$ bill, two envelopes contain a $\$20$ bill, and the
remaining two contain a $\$100$ bill. You randomly pick ...

I'm looking for a lower bound for binomial random variable $X$~ $B(n,p)$, where
$p=(1+\epsilon)/2$ for $\epsilon >0$.
I want to bound $Pr(X> n/2)$.
I know Suld's inequality, but it is good ...

The true odds against event $E$ are $r$ to $1$ and the payoff odds are $t$ to $1$. You bet an amount $B$:
You win $tB$ if the event occurs.
You lose $B$ if the event does not occur.
How do you ...

I'm trying to understand a good way to calculate the probability of landing on each box of the Monopoly board. In Monopoly you can move both rolling dice and when extra events happen (e.g., chance/...

Can u help me with this question pls
Assume that a gambler plays a fair game where he can win or lose 1 dollar in each round . His initial stock is 200 dollar. He decides a priory to stop gambling at ...

You roll twice with a six-sided, fair die. Let X be the sum of both throws, Y1 the result of the first throw and Y2 the result of the second.
How do I create Table of the probability function?

Let $A$ be some arbitrary finite set. (e.g. $\{a,b\}$).
Now let $$S=(A\times B)^n$$
Where $B$ is the set of all possible probability functions on $S$:
$$B=\left\{p\in S\to\mathbb R^+:\int_S p(s)ds=1\...

The number of text messages sent by Mario in an hour and the one sent by Ted are two independent Poisson random variables with mean 3. Let N be the total number of texts sent by the two guys between ...

The probability it rains on Wednesday this week is 40%, while the probability it rains on Thursday this week is 30%. However, it is twice more likely to also rain on Thursday, if it has already rained ...

Consider the following result from Wikipedia:
Let $(\Omega,\mathcal{F},P)$ be a probability space on which two sub $\sigma$-algebras $\mathcal{G}_1\subseteq\mathcal{G}_2\subseteq\mathcal{F}$ are ...

Let $\Sigma\in \mathbb R^{d,d}$ positive semi definit and $Z\sim N(0,\Sigma)$ a random vector in $\mathbb R^d$.
I want to show that $\Vert Z\Vert_2^2\overset{\text{d}}{\underset{\text{}}{=}}\sum_{k=...

On an aeroplane with 200 seats, if 5% of the passengers are sick, what is the probability that 2 sick passengers end up seated next to each other?
Had a few thoughts and approaches that led to ...

This is a stochastic process problem on a book:
A stochastic process $X(t) \equiv Y$ where $Y$ is a random variable and $E(Y)=a, D(Y) = \sigma^2$. Calculate the autocovariance of $X(t)$.
The answer ...

Let $X$ ~ $N\left(\begin{bmatrix}1\\-1\end{bmatrix}\right.$,$\left.\begin{bmatrix}9 & 1\\1 & 4\end{bmatrix}\right)$. Find distribution of $Y$ when $Y_1 = 2X_1 - X_2 + 4$, $Y_2 = 3X_1 + X_2 -3$....

I'm trying to find the pdf for a ring delta plus a complex Gaussian rv. I'm really not sure of the best way to approach this. The density of the ring delta is
$$
\frac{1}{2\pi r_0} \delta(r - r_0)
$$
...

(d) For nay nonnegative random variable X and $\alpha > 0$, Prove that
if $E[X^\alpha] < \infty$, then $\lim_{x \to \infty} x^\alpha P(|X| >
x ) = 0$
I tried to show this by using $E[X^\...

Let $(X,Y)$ ~ $N\left(\begin{bmatrix}2\\3\end{bmatrix}\right.$,$\left.\begin{bmatrix}7 & 2\\2 & 3\end{bmatrix}\right)$ . Find distribution of $T = 3X + 4Y$.
I know if $X$ ~ $N(\mu, \Sigma)$ ...

I have a function $Y=e^{-e^{a+bZ}}$, where $a$ and $b$ are constants and $Z$ is a standard normal random variable. I need to find $\mathbb{E}[Y]$. Is there even a closed-form solution to $\mathbb{E}[Y]...

In the classic counting problem where you have to give the number of possible ways to paint a cube with 6 different colors,
A well-known approach is constructive counting: fixing the first color to ...

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space (but the question is interesting for a general measure space).
Let $X_n$ be a sequence of random variables converging in $L^1(\Omega)$ to ...

Let there be a sequence of r.v.'s $\{X_{n}\}_{n\geq1}$. Show that if $\exists r \geq 1, \;\sum_{n=1}^{\infty}\mathbb{E}(\lvert X_{n}-X\rvert^{r}) < \infty$ then $X_{n} \xrightarrow[]{a.s.} X$
My ...

A urn contains 5 identical blue balls and 4 identical red balls. Taking 5 balls at random from the urn what is the probability that the number of blue balls be greater than the number of red balls?
...

I'm sitting with a task, in which I got the answer already. The task is the following:
"At a university, $15$ juniors and $20$ seniors volunteer to serve as a special committee that requires $8$ ...

I'm doing a course in probability and while calculating the characteristic function of the Gamma distribution my teacher did the following reasoning: (where $X\sim \mathcal{G}(r,\lambda)$)
By the ...

So given the joint pdf of $(X,Y) = 8xy$, as long as $0 < x < y < 1$ (and 0 otherwise). I got the conditional expectation to be $E(X|Y=y)=2y/3$ (if $0 < x < y <1$). Now I'm a little ...

I was given the excercise with solution:
And I am not finding the correct value.
My approach
$E[Y_1-Y_2]=\int^1_0\int^1_0(y_1-y_2)3y_1dy_1dy_2$
$=\int^1_0\int^1_0(3y_1^2-3y_1y_2)dy_1dy_2$
$=\int^...

Let $X$ have probability density of $f_X=xe^{-x}dx$ on $\mathbb{R}^+$ and $U$ the uniform distribution on $[0,1]$ independent of $X$. Define $Y_1=UX$ and $Y_2=(1-U)X$. The goal is now to show that $...

The application goes like this:
Let there be a random variable $X$, and $X_{1}, X_{2}, X_{3}$ random
variables representing a sample of $X$ that has the same distribution
as $X$ and that are ...

Exercising my mind a little bit with the problem and I've come to a dilemma. Picture the following scenario:
We have a standard dart board. We simulate the dart board and throw 100,000 darts at it, ...

I have 30% chance to have cavity gene. If I do have the gene, there is 61% chance that I will have at least one cavity over 1 year. If I don’t have the gene, there is 29% chance that I will have at ...

I'm reading on Hoeffding's covariance identity, the proof of which is neatly covered here, or, in a similar manner, in this MSE post, but I can't seem to fully understand the trick/property used there....

The stumbled upon following problem from Sheldon Ross's book:
Seven balls are randomly withdrawn from an urn that contains 12 red, 16 blue and 18 green balls. What is probability that either ...

I would like to know whether I have solved correctly the following problem, from Henk Tijms's Understanding Probability.
For a final exam, your professor gives you a list of 15 items to
study. He ...

Let's say there are $10$ orange balls and $3$ red balls in a bag.
And a person picks a ball at random and puts it on the side. So for example if he picked up an orange ball then there would be $9$ ...

The probability that Gao and D'artanyan hits a target are 1/4 and 2/5 respectively. If they shoot together, what is the probability that the target will be

I need to find de maximum likelihood estimator of $\theta$ for
$f(x)=\frac{1}{2}(1+\theta x)$, $-1 \leq x \leq 1$
I start with: $L(\theta)=f(x_1,\theta)f(x_2,\theta)\cdots f(x_n,\theta)$
$$L(\theta)=...

I'm working through a paper which can be found here. On page 13 assumption 3.2 we are given an assumption that for my application reduces to checking:
$$\int \sup_{x'\in X} \frac{\pi(x'|x)}{\pi(x)} d\...

There are $n$ students in the lesson and they're all tired,the time until a student falls asleep is exponentially distributed and independent from other students.On average a student stays awake until ...

I have the following problem in probability theory.
Suppose Alice and Bob agree to meet each other in the train station.
Alice is arriving between 16:00 and 17:00 (uniformly distributed) and stays ...

We have $(X_i)_{i \in \mathbb Z}$ iid random variables with $1\le X_i \le2$ almost surely.
We define $X(x,\omega) \equiv X_i (\omega)$ if $x\in [i,i+1[$ (so it is bounded almost surely) and, for $\...

Consider $X_1, X_2, X_3$ ... random variables i.i.d. such that $P(X_i=1)=p$ and $P(X_i=-1)=1-p$. Consider the random walk $(S_n)_{n\ge 0} $ with $S_0=0$ and for $n\ge 1 $, $S_n = \displaystyle\sum^{...

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