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 ...

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0answers
14 views

A variation of a combination and a permutation, I think?

The scenario is that 6 people have the option of choosing 8 doors and we want to know each door a person goes through. I have four/five questions based on this. 1) How many different ways can 6 ...
-1
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0answers
17 views

Combinatorics/Probability Question

You have a 20-letter alphabet and every sequence of length n can contain any of the alphabet at any position in the sequence independent of the other positions. You also have a test sequence and s ...
0
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2answers
14 views

Probability of a word where 2 letters do not follow each other

I have seven letters, say A, B, C, D, E, E, G. I have figured out how many distinct possible combinations I can have as $7!/2!$. My question is, how many of these will have the two E's separated? I ...
0
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1answer
6 views

Relative frequency

In a given scenario where two fair dice are thrown: what is the probability of the second roll being higher than the first? I can think of two ways to resolve this problem; 1- listing the possible ...
0
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0answers
8 views

Doob-Komogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
0
votes
1answer
13 views

Determining density involving scaled beta distribution

Suppose $Y \sim \mathrm{Beta}(2,1)$. If $X = \theta{Y}$ (for some $\theta > 0$) how do I determine the joint density $f(x, \theta)$? Edit: the density for $Z$ is $2z$. Would it be correct to say, ...
-1
votes
2answers
38 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
0
votes
2answers
19 views

How can I calculte the probability of $X$ with a Generlized Hyperbolic Distribution?

I would like to know how to calculate the probability of $X$ when I have fitted a Generalized Hyperbolic Distribution to my data set. The depth of my knowledge is basic t-tests and z-tests. I am ...
0
votes
1answer
19 views

Independent events and Kolmogorov

Suppose we have a probability space $(\Omega, \mathfrak{F}, P)$, and independent events $(E_n)_n$. Consider $$M_n = \sum_{k=1}^n I_{E_k}$$ Is it correct to say that by the Kolmogorov $01$ law $M_n$ ...
1
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1answer
23 views

Show that given $N$ iid variates $X_i$ uniform on (0,1), $P(\max(\{x_i\} > \frac{1}{2}\sum x_i)$ is $\frac{1}{( N-1)!}$

Given an ensemble of $N$ random uniform variates on $(0,1)$, the probability that the greatest variate exceeds the sum of all the other variates is $\frac{1}{(N-1)!}$. Is there any nice way to prove ...
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0answers
18 views

Probability helps to evaluate a sum

Let's consider a sum $$\sum_{n=0}^{m} \binom {n+m} {n} \cdot 2^{-n}$$. How does this sum can be evaluated, considering the topics about probability? One of the solutions is written at the "Concrete ...
1
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1answer
34 views

Let $X_{1},X_{2},…$ be iid random variables with distribution $P(X_{i}=x)=p$ if$ x=1$ , and $P(X_{i}=x)=1-p=q$ if $x=0$ [on hold]

The full question is here. I've done the part i) and part ii) But I'm not sure what to do with part iii). Any help is appreciated.
0
votes
1answer
24 views

Comparing sums of random variables

Consider $X_0,X_1\ldots,X_n$ mutually independent and $X_i \sim U(a_i,b_i)$. What is the probability that $\sum_{i=1}^n X_i<X_0$? Can you extend to mutually independent random variables with ...
1
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0answers
4 views

Meeting probability generalized: different wait times and number of meetings

I am looking to extend the problem of two people meeting for lunch, for example as found here: Chance of meeting in a bar However, I am trying to generalize this problem in two ways which, in ...
1
vote
1answer
33 views

Expectation how does $E[XY^2]=E[Y^2E[X|Y]]$?

Given random variables X and Y show that $E[XY^2]=E[Y^2E[X|Y]]$ For the case that $X$ and $Y^2$ are independent I have $$E(XY^2)=E(X)(E(Y^2)= E(E(X|Y))E(Y^2)=E(E(X|Y)Y^2)$$ but I'm sure about the ...
11
votes
9answers
1k views

Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...
0
votes
1answer
9 views

What's the Probability Of Non-Defective Mobiles?

In a shipment of 100 mobiles,6 are found defective.If Arpit buys two mobile from that shipment,what is probability of both-being non-defective ?
1
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1answer
45 views

Why is $P(X>r)=q^r$?

I was studying the geometric distribution when I came across a result that I did not understand. If $X$ follows a geometric distribution, where $p=$$probability$ $of$ $success$ and $q=$$probability$ ...
2
votes
2answers
22 views

Seating arrangement probabilites

Suppose that n people are seated in a random manner in a row of n theater seats. What is the probability that 2 particular people A and B will be seated next to each other? So I think that the number ...
0
votes
1answer
18 views

Multiple rolls of a one hundreded sided dice

Let's side I have a one hundred sided dice and a friend picks 8 random numbers from the dice. I understand that if I roll the dice one time, he has an 8% chance of getting the roll correct. My ...
0
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0answers
14 views

Zero variance Random variables with density

I found here the question: Can a random variable have a density function whose variance is $0$ ? I understood as a random variable which has a density. What is your opinion on what I ...
-3
votes
1answer
83 views

Can anybody tell me how to do this? [on hold]

A judge is 35% sure that xiao fang has committed a crime. Ahmad is a witness who knows whether Xiao Fang is innocent or guilty. however, Ahmad is Xiao Fang's friend and will lie with probabilty 0.25 ...
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0answers
6 views

Probability of exponential being smaller than a number given that its smaller than other exponential

We have X ~ exp(a) and Y ~ exp(b) independent random variables. Given a positive number n, we want to know: $P( Y < n | Y < X)$ Not sure how to proceed, tried using Bayes rule to calculate $P( ...
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0answers
27 views

Poisson Process Alterations

If we have a Poisson process of rate $\lambda$, do the following alterations still result in a Poisson process? 1) Deleting every alternate point 2) Inserting points at times $1, 2, 3, ...$ 3) ...
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votes
2answers
63 views

Comparing uniform random variables.

$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...
2
votes
2answers
35 views

Convergence a.e. of the series $\sum_{i=1}^{n^2} \frac{X_i}{n^2}$

Let $(X_n)_{n\geq 1}$ be independent random variables with expected value $m$ and $\sup_n Var(X_n)\leq K < \infty$, and they are uncorrelated. Then $1)$ $$\sum_{i=1}^n \frac{X_i}{n} $$ ...
-1
votes
0answers
25 views

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n ...
-1
votes
1answer
36 views

Probability of a nonnegative submartingale converging to zero [on hold]

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Thanks a lot. Some background ...
1
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2answers
27 views

In Markov chains a limit distribution is invariant

Suppose we have a Markov chain $(X_n)_{n \geq 0}$ with state space $S$. Suppose that $(\pi_i)_{i \in S}$ is a limit distribution. Then is $(\pi_i)_{i \in S}$ an invariant distribution ? I know the ...
0
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2answers
31 views

Probability of woman receiving positive mammogram and having cancer

The probability that a randomly selected US woman will have breast cancer in their lifetime is 0.12. Women over 40 are advised to have regular mammograms because early detection of breast cancer means ...
1
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3answers
35 views

Probability of number of people in car park at any given time

A building has 22 car spaces, each having a car parked within each spot in the morning. Each car is retrieved by its respective owner at some point (random time) between 7am and 9am (120minutes). Each ...
0
votes
1answer
24 views

Expected valued of Random sums about dice and jar problem

A six-sided die is rolled , and the number N on the uppermost face is recorded. From a Jar containing 10 tag numbered 1,2,,,,10 , we then select N tags at random without replacement. Let X be the ...
0
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0answers
9 views

Equation to denote a set based on probabilities

I have a set R with elements r. Each element has a certain probability P(r|X). Now a want a formal equation/notation for a new set E which contains the expected r elements when X happens. I can't ...
0
votes
2answers
17 views

How to calculate the probability of a die with a wild side?

So I have a 6-sided die with 5 different values in 5 of their sides. Its sixth side can be treated as any of the other 5 values. So my question revolves around which is the probability of getting any ...
0
votes
2answers
24 views

Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...
0
votes
1answer
3 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
0
votes
0answers
19 views

How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a sequence of natural numbers (i.e. this is a probability space of sequence of natural numbers sometimes used in ...
0
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0answers
17 views

Struggling with the notation of conditional expectation

Here's the question. I know the fomula of $E(X|Y=y)$ where y is a paticular number. But for this question, I really don't understand what's $E(X|Z)$ meaning. Help!
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0answers
16 views

Probability 2 balls in a bag [on hold]

consider an experience which consisit of drawing 2 balls with replacement from an urn containing 7 balls of which 3 are blue and 4 are yellow (i)what is the sample space (ii)define the events as a ...
1
vote
1answer
34 views

Find the value of $a$ and $b$ in $ F(x) = a + b \arcsin x $

Given $X$ is a continuous random variable and its probability distribution function is $$F(x)= \begin{cases} 0, & x < -1, \\ a+b\arcsin x, & -1 \le x < 1, \\ 1, & x \ge 1 ...
3
votes
2answers
45 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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0answers
8 views

MATLAB code based on a binomial random matrix [on hold]

I have a matrix, A, with $40000$ binomial random elements. I am trying to complete the following code and would appreciate help: I need to create $40000$ arrays $X_n$, where $X_n$ represents the ...
1
vote
1answer
29 views

Let n>=2, k>=2. The set of all k-element subsets partitioned into 4 classes: (i) class of subsets containing both 1 & 2, how many k-element subsets?

Sorry for the long title, I'm new here & not sure of the appropriate way to post long questions. The full question is: Let n>=2,k>=2. The set of all k-element subsets of [n] may be partitioned ...
1
vote
1answer
13 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
0
votes
1answer
18 views

Poisson Process problem, transform the possibility notation

Question: Suppose that a store opens at 0 pm and customers arrive according to a non-homogeneous poisson process ${N(t),t\ge0}$ with the intensity function $\lambda(t)=2t+1$ per hour. Let $S_3$ denote ...
1
vote
2answers
37 views

Multnomial coefficient combinatorics problem

The following problem: Ten diplomatic delegates are seated in a row. There are two specific seating requirements: 1) France and Britain are sat next to each other, and 2) the U.S. and Russia are ...
2
votes
0answers
23 views

How to calculate probability of users generating distributed events reaching n events per 15 minutes?

We have games & apps that connect to services such as Facebook and Twitter to fetch information. These services have various rate-limit caps that you cannot exceed - typically based on a 15 minute ...
1
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1answer
33 views

What is the probability that a random K-bit odd-number is prime?

Is it $e/K$? In an experiment that created 1000 random RSA-2048 key-pairs, 2000 random 1024-bit primes were created. It turned out that $727,709$ random candidates were generated, to create 2000 ...
4
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0answers
22 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
0
votes
1answer
22 views

Challenging Problem of Linear Permutation by H.C. Rajpoot

How many numbers are lying between 20045757087 & 87050752074 when all the 11-digit significant numbers, formed by permuting the digits 0, 0, 0, 2, 4, 5, 5, 7, 7, 7, 8 together, are arranged in ...