**59.765 probability questions.**

Question is as follow.
There is a bag. In the bag, there are $a$ red cubes and $b$ blue cubes. Assume that she knows exactly how many cubes for each of the colors before the draw. Mary is going to ...

In my textbook I have read:
Let $Y_1, Y_2, \dots, Y_n$ be a random sample of size $n$ from a normal distribution with mean $\mu$ and variance $\sigma^2$. Then
$$ \bar{Y} = \frac{1}{n} \sum_{i=1}^...

Let's say you have a source of random bit strings, which can generate a bit string of any length where each bit is independently set with fixed probability $p$, which I'll call my $p$-source.
Now ...

In infinite sets. Is there a difference between say, the natural numbers and the real numbers?
I know that there is a problem when trying to add the probabilities of hitting a certain real number ...

Please consider the problem below and my partial solution to it. Is it right so far? I do not know how to perform the integration and I am hoping somebody can point me in the right direction.
Thanks,
...

I want to estimate how many red balls in a box. Red, yellow, blue balls could be in the box. But I don't know how many of them are in the box.
What I did was randomly drawing 10 balls from the box ...

Consider an unfair coin flipped $n$ times. What is the probability of the following sequence occuring at least once? (for example)H T T HProbability of ...

The accepted answer here mentions that to sample points from the surface of a hypersphere uniformly, one can generate gaussians and normalize them. Many commentators also say that this is possible ...

Imagine that you are at $(0, 0)$ on a lattice grid, and your goal is
to reach $(4, 4)$. At each time step, you flip an unfair coin that comes up heads with probability $p$.
If you flip a heads you ...

Introduction to the problem
A set $C$ contains $c\in \mathbb{N}$ elements of three different kinds: There are $\alpha\in\mathbb{N}$ elements of kind $A$, $\beta\in\mathbb{N}$ elements of kind $B$, ...

Consider random variables $X_{i,j}$ with $j \in \{1,2\}$, $i \in \{0,1,\ldots\}$ so that they are i.i.d with $EX_{i,j} = 0$ and $ \operatorname{Var}(X_{i,j}) = \sigma^2$ and assume all moments exists. ...

I'm trying to understand why the hypothesis mutually independent is not necessary in the following theorem
In the later section, Linearity of expectation, there is an example I don't understand
But ...

My question is about how to get $\int^u_0\int^{u-y_2}_0$ and $\int^1_{u-1}\int^1_{u-y_2}$.
How to find the sum of two uniform random variables by method of distribution? (without using convolution)
...

A gambler plays a fair game where he can win or lose $\$1$ in each round. His initial stock is $\$200$. He decides a priori to stop gambling at the moment when he either has $\$500$ or $\$0$ in his ...

Let $X_i$, $i\geq 1$, be independent and identically distributed random variables having the uniform distribution over $(0,1)$. Let $X$ be defined as $X=\sum_{i=1}^{N}X_i$, where $N$ is an unknown ...

I am confused as to which formula I should use in the following situation.
Say I have a series of independent events which can happen with probability $p$ (very small positive). I want to know how ...

I tried to solve a problem two different ways and I got different results.
Let $( X_i )_{i \in \mathbb{N}}$ be a series of independent, identically distributed random variables, with $\mathbb{E}[X_i] ...

In Esser, Kübler, May (2017), Chernoff's bounds are used to check that the Hamming weight of something could be part of a Binomial distribution.
I need to change the upper bound, as we will be ...

In the constructive form of the LLL we bound the expected time using a weight function
$x:\mathcal{A}\rightarrow (0,1)$
that satisfies
$\forall A\in \mathcal{A},\Pr [A] \leq x(A) \prod _{B\in \...

I am trying to prove the following statemt:
Let $(\Omega, \mathcal A, P)$ be a probability space. Let $(Z_n)_{n\in \mathbb N}$ be i.i.d. random variables with $Z_1 \in \mathcal L^1$. Let $\theta \...

I recently started to learn about Markov Chains and had a problem regarding the expected time to absorption:
Problem:
Markov has an untrained mouse that he place in a maze. The mouse may move ...

The question that I have:
"Let $X$ be a distribution over $\mathbb{N}$ with mass
$P(X = i) = \frac{\alpha}{2^i}$
If $\alpha \in \mathbb{R}$, find $\alpha$ and $E[X]$"
My logic goes as follows:
...

I am trying to solve exercise 2.8 of the book "Pattern Recognition and Machine Learning by C. M. Bishop, but I get stuck. The question is as follows.
Consider two variables $x$ and $y$ with joint ...

I have some population data with a known mean and median for the entire population and I am trying to calculate the probability of the distribution below a value. I've spent a long time searching and ...

An urn problem
An urn $C$ contains $c\in \mathbb{N}$ elements of three different kinds: There are $\alpha\in\mathbb{N}$ elements of kind $A$, $\beta\in\mathbb{N}$ elements of kind $B$, and $\gamma\in\...

Let $X_1,\cdots,X_n$ be sequence of positive, iid random variables such that $\mathbb{E} X_1 <\infty$. How can I show that
$$\frac{1}{n}\max\left(X_1,X_2,\cdots,X_n\right)\to 0 \text{ in ...

I was given the follwoing question with solution:
I however do not understand how they produced their answer.
First, what general formula was used to produced the value $p(y)=\int...d\lambda$?
...

For example, if we have an Ace of spades, the next card cannot be an Ace nor spades.
Edit: Assuming that we pick one particular shuffle amongst all possible ones

$f(x, y) = 6x$ for $ 0 \leq x \leq y \leq 1$
How do I calculate the following probability ? I haven't done double integrals in years and can't understand how this works.
$P(X < 1/2, Y < 1/2)$

$f(x, y) = 6x$ for $ 0 \leq x \leq y \leq 1$
How do I calculate the marginal pdfs in this equation? I haven't done double integrals in years and can't understand how this works.

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.
Hints only please!
This is a confusing worded-problem.
We could ...

Recently, I see some properties from conditional independence wiki page https://en.wikipedia.org/wiki/Conditional_independence
I don't quite understand the properties of "Rules of conditional ...

Considering the random variables $X_1,\ldots,X_n$, i.i.d, with Cauchy distribution, and the random variable $Y_n=\frac{X_1+\cdots+X_n}{n}$
Determinate the characteristic function of $Y_n$ and ...

Suppose the random variable $T$ which represents the time needed for one person to travel from city A to city B ( in minutes). $T$ is normally distributed with mean $60$ minutes and variance $20$ ...

The exercise: A group of N friends sits around a table shaped as a regular polygon with N sides, one person on each side. Everyone tosses a fair coin once and a person is called positive if she and ...

A stochastic process $(X_{n})_{n \in \mathbb{N}}$ on the state space $S$ is said to be a markov chain, if it satisfies the markov propery, i.e. for all $n \in \mathbb{N}$ and $i_{0},...,i_{n+1} \in S$ ...

Let $X_1$ and $X_2$ be i.i.d. normal.
The question is:
Can we find the joint distribution of a pair
\begin{align}(U_1,U_2)=(\max(X_1,X_2), \min(X_1,X_2)).
\end{align}
What I did
Note that ...

Say you were to “host” a single elimination tournament of 8 dice, each with a different number of sides: 4, 6, 8, 10, 12, 20, 24, and 30. In each round, the dice that rolls the highest wins (re-roll ...

Let $A=\{1,2,3,...,1000\}$. Find the probability that a randomly chosen element of $A$ would be a multiple of $2$ OR $3$.
My Attempt:
There are $500$ numbers multiples of $2$($\frac {1000}2$). And$\...

Are there any well known canonical forms for stochastic matrices where both the transform vectors as well as the $\bf C$ matrix are always non-negative ( all vector entries $\geq 0$ )? And for what ...

Q. A binary operation is chosen at random from the set of all binary operations on a set $A$ containing $n$ elements. The probability that the binary operation is commutative is?

$1$.$X_n$ is iid Gaussian process and $U_n$ is iid binary random process with $Pr${$U_n=-1$}=$Pr${$U_n=1$}$=0.5$.$X_n$ and $U_n$ are independent
$2.Y_n=X_n+U_n$,and $\hat U_n=Q(Y_n),$where $ Q(Y_n)=\...

Assume we have $n$ red balls, $n$ green balls, and unknown number of white balls.
We select each ball to a set with probability $p=\frac{1}{n}$ and not choosing it with probability $1-p=1-\frac{1}{n}$...

Let $\mathcal{Z}\subset \mathbb{R}$ be a set at most countable. Let $X,Y,Z,\xi$ random variables with the following properties:
$X,Y,Z$ have value in $\mathcal{Z}$
$\xi\ $ has a Bernoulli ...

Suppose I have a line segment of length $L$. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?

I feel like my reasoning and solution for this question are satisfactory, but it was a multiple choice question with explicit decimals given for the options and the expression I arrived at for a ...

Let ${X_i}$ be independent, identically distributed, random variables each with mean $M$ and variance $\sigma^2$. Let $Y(z)$ be the number of these random variables we need to add together to exceed z,...

- probability-theory
- probability-distributions
- statistics
- combinatorics
- random-variables
- expectation
- stochastic-processes
- normal-distribution
- measure-theory
- markov-chains
- discrete-mathematics
- conditional-expectation
- dice
- integration
- calculus
- real-analysis
- permutations
- combinations
- convergence
- uniform-distribution
- bayes-theorem
- bayesian
- poisson-distribution
- statistical-inference
- random-walk

- Are crime rates up by 10% in Germany?
- Prove A is not invertible
- Is this a real display in the United States Holocaust Museum?
- Gender transition as a starting faculty member: a terrible idea?
- How fast can I flood the Netherlands entirely and permanently?
- Is there any evidence of armies enrolling women in fighting roles in significant number in ancient history and middle ages?
- Non-subjective definition of "terrorist", or widely used equivalent term?
- Why was the Council of Nicea held in Nicea rather than Rome or Constantinople?
- How to deal with supervisor who required me to share all my work with him?
- Let's design a digit mosaic
- Intuitively, what exactly does the ellipse equation mean?
- It's now safe to turn off your computer
- Did "user" have positive or negative connotation in 1950?
- How to avoid that anytime(<numeric>) "updates by reference"?
- An ellipse through 12 points related to Golden ratio
- Can a planet harbor plants of different colors without one pigment outcompeting the others?
- How can I convey to my technologically inept parents that my IT job is mentally draining
- Can secret GET requests be brute forced?
- How to get past the "mystique" of Maths
- Why would we use direct current instead of alternating current?
- Why solder mask are not applied to RF PCBs
- How do we recover units of force from units of gravitational potential?
- What are these Russian characters that aren't in the alphabet in my learning material?
- Floor function of a real number