All questions with tag [math: probability]


Card probability problem

Possible Duplicate: ¢ I located the adhering to trouble in Rosen is Discrete Mathematics and also Its Applications 6th ed. : There are 3 cards in a box. Both sides of one card are black, both sides of one card are red, and also the 3rd card has one black side and also one red side. We select a card randomly and also observe just one side. ...
2022-07-25 17:47:14

Probability of Heads in a coin

I was asking yourself, if you turn a reasonable coin $5$ times, whether you can compute the probability of accessing the very least one head is computed like this: You can do the enhance of accessing the very least one head which is TTTTT: $\dfrac1{2^5} =\dfrac1{32}$ Then you do $$1-\frac1{32}= \frac{31}{32}\;,$$ to make sure that is the oppor...
2022-07-25 17:45:56

Handling dropouts in a round-robin tournament.

In a rounded - robin event (where a gamer versus every various other in the event subsequently), do gamers that quit of the event after a couple of rounds influence the probability of winning for those that bet them throughout the training course of the event until now (in cases that some rivals shed to this gamer while others won)? If they do ...
2022-07-25 17:45:48

Expectation of Random Variable with even Probability Density Function

By Definition of Expectation of Random Variable: $$ E(X)= \int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$ Now if the pdf $f_X(x)$ is Even we know that $E(X)=0$ (Ofcourse if integral Converges, i.e, Lets exclude cases like Cauchy Random Variable) Is the Converse True, i.e., is there a Random Variable $X$ whose pdf is Neither Even-Nor Odd, such that $E(...
2022-07-25 17:45:31

Two-sided Chebyshev inequality for event not symmetric around the mean?

Let $X$ be a random variable with finite expected value $μ$ and non-zero variance $σ^2$. Then for any real number $k > 0$, two-sided Chebyshev inequality states $$ \Pr(|X-\mu|\geq k\sigma) \leq \frac{1}{k^2}. $$ I saw a paper applies two-sided Chebyshev inequality to $\Pr(|X|\geq b)$ and got an upper bound $Var(X)/b^2$, when $X$...
2022-07-25 17:44:46

The distribution of the sum of two independent exponential distributions

I am trying to calculate the distribution of the sum of two independent log-uniform distributions but something doesn't add up. Suppose $a \sim \mathrm{uni}(0,1)$ and $b \sim \mathrm{uni}(0,1)$. Thus, $u=\log(a)$ has an exponential distribution of the form $e^u$, which is defined for values for which $u<0$ (the same applies to $v=\log(b)$...
2022-07-25 17:43:22

Is this function $f(x) = 2 - \Phi(a-x) - \Phi(a)$ log-concave?

I would like to verify this function $f:\mathbb{R} \to \mathbb{R}$ to be log-concave or determine the region where it is: $$ f(x) = 2 - \Phi(a-x) - \Phi(a) $$ where $a > 0$ is a constant, and $\Phi(x) := \frac12\left[1 + \operatorname{erf}\left( \frac{x}{\sqrt{2}}\right)\right] $, i.e. the CDF of the standard normal distribution. My att...
2022-07-25 17:42:53

Sum of 3 loaded dice

I am given 3 loaded dice $D_1$, $D_2$ and $D_3$ and their probability tables $P(D_i = k), 1 \leq k \leq 6$. I ought to write an algorithm that computes $P(\text{Sum} \mid D_1)$, the sum of all three dice conditioned on the value of the first die. My intuition lead to the following solution: For each possible sum of two dice ($2,\ldots,12$) I col...
2022-07-25 17:42:17

dice probability issue

In the video game Settlers of Catan, territories/tiles are each (properly) arbitrarily appointed a number from 2 - 12. When that number is rolled as the amount of 2 dice, the floor tile creates sources for the gamer or gamers that have actually resolved it (practically: along it). A crucial element of the video game is taking into consideration ...
2022-07-25 17:41:18

Probability involving unique group combinations

If I have $30$ objects and $5$ buckets that each hold $6$ objects, how many times could I put the objects into the buckets without an object being in the bucket with an object it has previously been grouped with? So, for each round you would empty all of the objects from the buckets and place them into buckets again (they could be in the same b...
2022-07-25 17:40:05

Joint Probability $P(X,Y,Z) = P(Y,X,Z)$

Does the order of variables in the joint probability $P(X,Y,\dots)$ have any kind of effects on the declaration of joint probability? Concrete, is: $$P(X,Y,Z) = P(Y,X,Z)$$ To my mind, plainly this is proper, can a person ensure me?
2022-07-25 17:40:02

Probability Questions

I need help with part (1) of this problem and confirmation I've done the right thing for (2). 1) A survey carried out by a firm found 60% of clients buy products every month and 20% buy high-tech products. Of those who buy products every month, 30% buy high-tech products. (i) Are the events 'buy products every month' and 'buy high-tech products'...
2022-07-25 17:22:24

Battleship probability matrix

Consider a 10 x 10 Battleship grid that conceals a solitary ship of size = 3. This ship can be positioned up and down or flat in any one of the 100 cells. The trouble is to get the 10 x 10 probability matrix that maps the most effective cells to fire (greater probability to strike the ship). My first suggestion was to loop via each cell, and al...
2022-07-25 17:22:19

MA process ACF proof - don't understand it

I've got the proof but I don't understand a small detail. As you know for an MA process: $X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$ where $Z_n$ is WGN (pure Gaussian random process). Then the ACF is: $\gamma(k) = Cov(\sum _{i=0} ^q \beta_i Z_{n-i}, \sum _{j=0} ^q \beta_j Z_{n-j + k}) = \sum _{i=0} ^q \sum _{j=0} ^q \beta_i \beta_j Cov(Z_{n-i}, Z_{n-...
2022-07-25 17:20:40

$\sigma$-algebra generated by Brownian motion

Let $(B)_{t \geq 0}$ be a standard Brownian motion. Then $B$ is adapted to its natural filtration $(\mathcal{F}^B_t)_{t\geq 0}$. Often, we want to consider a slightly bigger filtration, ones satisfying the right-continuity condition. In that case, we define for every $t \geq 0$: \begin{align*} \mathcal{F}_t^+ = \bigcap_{s > t} \mathcal{F}...
2022-07-25 17:20:40

Strange Patience Game

I review this video game as a child, yet my mathematics was never ever approximately addressing it: The rating begins at absolutely no. Take a mixed pack of cards and also maintain dealing face up till you get to the first Ace, at which ball game comes to be 1. Bargain on till you get to the next 2, at which ball game comes to be 2, although yo...
2022-07-25 17:17:29

Tail bound for sum of product Normal-Bernoulli random variables

Let $Z \sim \pi N(\mu, \sigma^2) + (1-\pi)\delta_0$ and also $z_i \sim Z$ are iid for $i=1,\ldots,n$. I would love to get an outcome of the kind $$ P[n^{-1}\sum_i z_i - \pi\mu > \epsilon]\leq\exp(-c n \epsilon^2) $$ with a specific constant c. I calculated ${\mathbb E}[\exp(t(Z - \pi\mu)] = \exp(-t\pi\mu)[1-\pi + \pi\exp(\mu t + \frac{\s...
2022-07-25 17:16:33

Conditional Probability of Jointly Distributed Random Variables.

The arbitrary variables X and also Y are collectively dispersed according to the pdf $f_{X,Y} (x,y)=x+y, 0 <x<1, 0<y<1$. Locate $P(Y>1/3|X=1/2)$ Since this is a conditional probability, can not I simply write $P(Y>1/3|X=1/2)=P(Y>1/3 , X=1/2)*P(X=1/2)=0$ given that $P(X=1/2)=0$ - due to the fact ...
2022-07-25 17:16:08

Weighted average of two frequencies

This need to be actually straightforward yet I'm obtaining stuck and also I'm possibly exceptionally foolish. I recognize that an equipment obtains 2 sort of parts: Type 1 - > with regularity 50 weekly, every one is refined for 20 mins Type 2 - > with regularity 100 weekly, every one is refined for 8 mins It was informed me to take into...
2022-07-25 17:15:49

Lower bound on a minimum of maximum of a sequence of standard normal random variables

Let $X = (x_{ij}) \in \mathbb{R}^{n \times p}$ be a matrix with independent $N(0,1)$ access. We understand that $\max_j x_{ij} < \sqrt{2\log(p/\delta)}$ with probability at the very least $1-\delta$. I would love to get a lower bound for $\min_i (\max_j x_{ij})$ that accepts probability at the very least $1-\delta$. Could someone indica...
2022-07-25 17:14:43