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Questions tagged [probability]

This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

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12 views

On distribution of the time between two consecutive events (either arrival or departure) in an M/M/1 queueing system

For an M/M/1 queueing system, distribution of the time ($t_e$) between two consecutive events (either arrival or departure) can be derived as follows with the independent assumption, $$F(t_e\ge t)=F(...
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1answer
22 views

MLE for mean in geometric distribution

i have a random sample of n observations from geometric distribution pdf $p q^x-1$ but i m not able to find the 1) maximum likelihood estimator of the mean , 2) MLE of P using exponential family ...
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2answers
31 views

How to prove a state is recurrent in a Markov Chain?

Examine the nature of the states of the Markov chain having transition probability matrix link Here is my approach so far. link What I did was follow the technique mentioned here. link However, I ...
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1answer
12 views

Expected value of second run in a sequence of coin tosses

Let $(X_n)_{n\in\mathbb{N}}$ be independent random variables that are equal to $1$ with probability $p$ and to $0$ with probability $q = 1-p$. A run is a sequence $(X_{k}, \cdots, X_{k+l})$ where $k,l\...
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1answer
28 views

CLT for inner product of Hilbert space valued random variables

Let $\mathcal{H}$ be a Hilbert space over $\mathbb{R}$ with inner product $\langle \cdot , \cdot \rangle$ and let $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ be i.i.d sequences of random ...
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0answers
34 views

“multiple” of markov chain properties

Let $A_n$ be a markov chain with transition matrix $P$ and $B$ be a chain defined as $B_n = A_{mn}$ for some positive integer $m$. First, I was able to show already that $B_n$ is also a markov chain ...
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1answer
15 views

Statistically Ranking Data

We are sampling the rates of $n$ elevators rising up floors in order to see which elevators are the fastest, and which are the slowest. Say we have $k$ time samples from each elevator, $S_1, S_2, ... ,...
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1answer
24 views

How to find something that is 1 standard deviation of the mean value?

I am having difficulty with one particular item in my homework. Its in the section of the text on "Continuous Random Variables" The homework question reads as follows: "Time headway" in traffic ...
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0answers
35 views

Choosing my most preferred card from a set of n cards.

I have $n$ cards, however, I like only 1 card the most out of all the $n$ cards and that card is my favourite. I consider the cards one by one, giving each an integer score, where the higher the score ...
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1answer
16 views

Rate of convergence precise meaning/definition

In the context of mathematical statistics, people often say MLE's rate of convergence is $\sqrt{n}$ as $\sqrt{n}(\hat{\theta}_n-\theta_0) \to N(0,V)$ in distribution. But is there any official ...
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1answer
25 views

Bounding the probability of a set of non independent events

Suppose I have the following events $A_0, A_1, ... A_N$. These events are not independent. I want to know the probability of the event $A_0 \cap A_1 \cap ... \cap A_N$ occurring. However, I don't ...
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0answers
62 views

Show $P(T < t) =(1-e^{-2\alpha t})^{2}$

A system consists of four components that are similar and work independently, say $A,B,C,D$. To operate properly, it is necessary that $(A$ and $B)$ or $(C$ and $D)$ are functioning properly. Define ...
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0answers
13 views

Establishing difficulty of Mahjong Solitaire game layouts

I'd like to know if it's possible to calculate the difficulty of Mahjong Solitaire layouts and show it's difficulty as a factor that compares with other layouts - and what that way of calculating ...
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0answers
30 views

Is the null hypothesis test true?

A friend of yours offers to play a little game with you. The rule of the game is very simple: After a toss of a coin, if the coin lands on its head, he gives you £10; if it lands on its tail, you give ...
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1answer
63 views

Probability of at least $3$ people sharing the same birthday in a group of $n$ people

Disclaimer: I've seen posts with good answers for the case of "at least $2$ people", or "exactly $2$ people". Posts with "at least $k$ people" usually suppose that the mutual birthday is a fixed day (...
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0answers
18 views

Stopping moment regarding the ratio of black balls to all balls.

The urn contains at start one black ball, one white ball and one green ball. In the next steps, we randomly pick a ball from the urn and put it back with one additional ball coloured as the one we ...
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1answer
55 views

If $X, Y$ ~ $\exp{\alpha}$ determine the PDF of $\frac {X}{X+Y}$

Let $X,Y$ be independent random variables the are distributed $\exp{(\alpha)}$ whereby $\alpha > 0$ Determine the Distribution Density of $\frac {X}{X+Y}$ My idea: I believe it is too simple ...
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0answers
26 views

simulation/ probabilty

Suppose a population control plan for the country of Transylvania allows parents to have at most four children each, and they must stop having children when they get two girls. Explain how to ...
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3answers
63 views

Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the distribution function of $Z=X+Y$ - how to break into cases?

Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the distribution function of $Z=X+Y$ - how to break into cases? I first found the density functions: $$ f_x(t) =\begin{cases} 1 &...
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1answer
32 views

$P(X>0|X+Y)$ where X,Y have poisson distribution

X,Y both have poisson distributtion with parameters $\lambda$ $\nu$ accordingly: I have to calculate $$P(X>0|X+Y)$$, my first question is, does this mean I have to calculate it for every $j$ where $...
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0answers
41 views

Distinguishing two channels with known probability distributions

Let $p(\cdot | \cdot)$ and $q(\cdot | \cdot)$ be two known channels from alphabet $X$ to $Y$. What is the optimal strategy to distinguish between them if you are allowed to use the channel only once, ...
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0answers
33 views

If $X_1, \ldots, X_N$ are NOT identically distributed nor independent, how would the $E(X)$ be different than what we have normally?

Suppose $X_1, \ldots, X_N$ are NOT identically distributed nor independent. Now, suppose I wanted to compute expectations, such as: $$ E(X|Z) $$ where Z is another variable. Does this change ...
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0answers
12 views

If I have a random variable $X$ that follows a distribution identically (but not independently), $X \overset{id}{\sim} F$, why is this non-parametric?

Suppose I have a random variable $X$ that follows a distribution identically (but not independently), $X \overset{id}{\sim} F$. This is usually meant to say that $X$ has a non-parametric specification....
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1answer
38 views

Variant on Gambler's ruin problem?

Suppose I started with $W_1=1$ and decided on a ratio $0<\alpha<1$ such that I invest $\alpha W_n$ of my earning in the next round. I either lose my investment, or win it with probability $p=9/...
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2answers
37 views

Probability that a burglarly is ongoing?

A shop is equipped with an alarm system which in case of burglary alerts police with probability $0.99$. During a night without a burglary, a false alarm is set off with probability $0.002$. The ...
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1answer
33 views

Sample space of drawing 3 balls from a bag of 6 black balls and 5 while balls.

If 3 balls are randomly drawn from an urn containing 6 white and 5 black balls, what is the probability that one of the drawn balls is white and the other two black? 1) What is the sample space for ...
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2answers
35 views

Probability that 5 heads occur in 8 tosses [on hold]

Suppose that you toss a coin until, for example, 5 heads occur. What would be the probability that it takes 8 tosses?
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0answers
12 views

Distribution of color observation in Polya-like urn model

Consider an urn containing balls of $K$ different colors with $a_i$ balls of colour $i$. According to Wikipedia, if we sample $n$ balls in a Polya's fashion, the distribution of the observed colors ...
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0answers
12 views

Find pdf of rectangular signal with random variable as width

I have a random process which is $\pi_p(t)$ which is the rectangular function in the interval $[0,p]$. $p$ is a uniformly distributed random variable over interval $[0,10]$. I want to find the pdf of ...
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2answers
46 views

What's wrong with my calculation? $\mathrm E[X^3]$ where $X\sim N(0,1)$

Let $X\sim N(0,1)$, compute $\mathrm E[X^3]$: My attempt: $$ \mathrm E[X^3]=\int_{-\infty}^{\infty}x^3\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\mathrm d x \\ x^2=t \Rightarrow \mathrm dx=\frac{\mathrm ...
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0answers
51 views

Asymptotics In Probability Theory Help

I have a problem with a random walk I'm trying to work with. Suppose I have a random walk $$S_n = \xi_1 + ...+\xi_n, n \geq 1, S_0= 0$$ with i.i.d. increments $\{\xi_n\}$ with common distribution ...
4
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1answer
38 views

Show that $p_{n} \geq 1- \exp{(-n(n-1)/730)}$

On the issue of the birthday paradox,Let $p_{n}$ be the probability that in a class of $n$ at least $2$ have a their birthdays on the same day (exclude $29$ Feb). Use the inequality $1-x \leq e^{-x}$ ...
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1answer
58 views

Sitting on a bench and getting up without apologising

There's a long table and behind it - a long bench. The only way to sit and get up and leave from the bench is through one side. Behind the table, there are 14 students and they're writing an exam. ...
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3answers
35 views

Probability of a pair

Let there be four cards (2 queens and 2 kings). You draw two cards. What is the probability that the cards are a pair? Is there a difference in probability if you draw two cards together (in which ...
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1answer
25 views

Multinoulli Distribution Explanation

I know about multinoulli distribution, but I found a different explanation in a book that I have been reading and I didn't quite get it. It says: The multinoulli, or categorical, distribution is a ...
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1answer
36 views

Difficulty finding average in a Poisson distribution Question

The question states: A very large number of Aedes mosquito larvae are on the surface of a river, scattered randomly so that the number of larvae is, on average, 3 per square cm. For a study of the ...
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1answer
30 views

Finding $P(X < cY)$ when $X$ and $Y$ are iid random variables [on hold]

If I know that $X$ and $Y$ are iid random variables, is there are special formula for $P(X < cY)$ where $c$ is some nonzero scalar?
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2answers
10 views

Finding the conditional distribution of the uniform distribution

Let $U$ denote a random variable uniformly distributed over $(0,1)$. Compute the conditional distribution of $U$ given that $$(a)\,\,\,U>a\\ (b)\,\,U<a$$ where $0<a<1.$ For $(a)$ I ...
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1answer
43 views

Solving $P( Y < 0.5 | X > 0.5)$

On the following page $P(Y<.5 | X >.5), P(Y>2X), P(.5<X+Y<1.5)$ with Joint probability density function. it gives the following problem... "Let X and Y have the joint probability ...
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0answers
24 views

Steps to calculating this integral needed to find an expected value

Find $$\int_0^1 \int_0^1 e^{t_1 x}e^{t_2 y } \frac{2}{5}(2x+3y) dx dy$$ I've been working on this for a day and can't figure out how to calculate it. Additional information: $$M_{XY}(t_1,t_2)=E[e^{...
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5answers
26 views

Probability of Angela and Blake sit next to each other if 10 friends are randomly assigned seats?

A group of 10 friends attend a concert together. I have found out that having 10 seats there are 10! ways for the friends to be seated. But my question is how do i figure out the probability that 2 of ...
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2answers
21 views

Probability that x is true given some amount of successful trials

I am having a hard time wrapping my head around this question: Suppose that I don't know if I am playing with a fair set of dice or not, but I know my probability of having the loaded dice is .38 if I ...
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1answer
27 views

Show that $n^{\alpha}X_{n} \xrightarrow{n \to \infty} +\infty$ almost surely

Let $(X_{n})_{n}$ be a sequence of random variables that are identically distributed on $\mathcal{U}(0,1)$. Furthermore, let $\alpha > 1$. Show that $n^{\alpha}X_{n} \xrightarrow{n \to \infty} +\...
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0answers
18 views

Determinant of Jacobian equivalent to absolute of derivative

In this report, Equation 10 it is claimed that we can make the jump from $$ p(y) = p(f^{-1}(y)) \cdot \left| \frac{d f^{-1}}{dy} \right| $$ to: $$ p(\mathbf{y}) = p(f^{-1}(\mathbf{y})) \cdot \left| \...
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2answers
19 views

Probability graph will have some nodes with full-mesh connectivity.

There is a graph with $n$ nodes. This makes for $n \choose 2$ edges. We know for sure $l$ of these $n \choose 2$ edges are present and the rest are absent. What is the probability that for some $k<\...
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2answers
34 views

How do I model the probability that a straw is missing and another is in perfect condition

In a pack of six chocolate drinks every carton is supposed to have a straw, but it is missing with probability $1/3$, with probability $1/3$ it is broken, with probability $1/3$ it is in perfect ...
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2answers
20 views

How many diamonds would you expect to be drawn if 9 cards are randomly selected?

Question: a) How many diamonds would you expect to be drawn if 9 cards are randomly selected from the deck? b) How many cards that are multiples of 3 cards would you expect to be drawn if 4 cards ...
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1answer
9 views

Probability of incorrect classification in majority vote

In a supervised learning problem, I have 11 models, which have an error rate of 0.2 What is the probability of an incorrect prediction? equation I have the equation (n/k)p^k(1-p)^n-k but something ...
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0answers
43 views

Probability of winning the lottery in Turkey

Lottery Administration prints 10 million tickets with 7-digit numbers in Turkey. Of these tickets, 8.800.000 are quarter-price, 700.000 half-price and 500.000 full-price tickets. The quarter-price ...
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2answers
63 views

A counting problem of probability

Problem $:$ How many ways are there to split a dozen people into $3$ teams , where one team has $2$ people and the other two teams have $5$ people each? My try, Pick $2$ guys from $12$ guys which ...