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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Expectation of the conditional density

What is the difference between E[$X_1$|$X_n$ = $x_n$] and E[$X_1$|$X_n$]? I have found the first one, by integrating x*$f_{X_{(1)}|X_{(n)} = x_{(n)}}$ (x). If anyone has pointers for finding ...
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3answers
78 views

Central Limit Theorem and Mean time between failures

I was reading up about RAID, and the text said: Suppose that the mean time to failure of a single disk is $100000$ hours. Then the mean time to failure of some disk in an array of 100 disks ...
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0answers
12 views

Probability Help with finding mean and variance of estimators [on hold]

Consider a random sample, $X_{1} , X_{2} , . . . , X_{n} (n > 2)$, from a distribution with mean μ and variance $σ^{2}$. You may assume that $σ^{2}$ is known. Three estimators are proposed for μ:$$ ...
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1answer
29 views

Finding PDF of function of a random variable

Suppose $X$ has PDF: $f_X (x)= \lambda e^{-\lambda(x+2)}$ , for $x \ge-2$ $f_X(x)=0$ , for $x <-2$ Determine the PDF of $Y = X^2$. I am stuck because for $-2\le X \le 2$, $0\le Y \le 4$, and I ...
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1answer
28 views

Special dice generating non-decreasing sequence

Suppose that, when rolled for the first time, a special 6-sided dice shows $1,\ldots, 6$ with probability $\frac{1}{6}$ each, and then, upon rerolling, shows with equal probability a number greater or ...
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1answer
16 views

For a sequence of i.i.d. (Bernoulli ?) RV we have for the partial sums $S_{n+m}-S_n=m$ i.o. almost surely

Problem: Given a sequence $(X_n)_{n \geq 1}$ of i.i.d. RV and $P(X_1=1)=P(X_1=-1)=1/2$ we have for $m \geq 2k+1 \in \mathbb{N}$ for the partial sums $S_{n+m}-S_n=m$ i.o. a.s. My approach: I want ...
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4answers
48 views

Expected Value help

A fair coin is tossed. If a head occurs 1 die is rolled, if a tail occurs 2 dice are rolled. Let X be the total on the die or dice. What is E[X]? To be honest, I don't get this. The answer was ...
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0answers
16 views

Success runs in dependent trials

There are 260 business days in a year. We have 54 employees. Each employee is required to bring donuts twice a year on different days. Each employee chooses the two days at random, independently of ...
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1answer
20 views

Drawing exactly $r$ red, $g$ yellow and $b$ blue balls out of an urn

In an urn, let there be $U \in \mathbb{N}$ balls. Of these balls, $R$ are red, $G$ are yellow and $B$ are blue, and there are no other colors than these in the urn. (So, $R + G + B = U$.) Now, without ...
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0answers
13 views

Hammersley–Clifford theorem

I'm looking to see an example of a use of the theorem. It states that the joint density of a vector is proportional to a product of functions over the maximal cliques of the associated graph. can ...
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1answer
17 views

Probability space for zebras and their number of stripes

On a trip to Africa the researcher Alison notices that zebras with an even amount of stripes have double the probability to be seen than zebras with an odd amount of stripes. Let $E_n$ denote the ...
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1answer
18 views

Joint probability with housing stock data

I have a question about probability. Let’s say we have 100 homes of different ages, types and insulation levels, distributed as per the table below. Housing stock data How do I determine how many ...
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0answers
12 views

Set interpretation - topology vs probability

Consider the sequence of i.i.d. distributed random variables $(X_i)_{i\geq1}$ on $\mathbb{Z}^d$. We define the following norm $I(x)=\mid x\Gamma^{-1}x\mid$, where $\Gamma$ denote the covariance matrix ...
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1answer
16 views

Help with the poisson process and jackson networks.

I'm posting the following question from my notes as an image because it has a diagram within it. It's from my lecture notes. I start to get confused when $\delta $ is choses to be much smaller ...
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0answers
28 views

How to prove these two random vectors has the same distribution?

I find this problem when reading a paper. The author seemed to think it is trivial so did not list it as a lemma or something. The question is : $\tilde{u}$ is a random unit vector in $R^D$, $u$ is a ...
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2answers
27 views

The probability of a student speaking spanish is $30\%$. If we select $3$, what are the chances of at least one of them speaking Spanish?

In a school the probability of a student speaking Spanish is $30\%$. If we select $3$ random students, what are the chances of at least one of them speaking Spanish? So, I saw this question and ...
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1answer
27 views

Expected value question.

This is a question from my lecture notes. """ Persons arrive at a copy machine according to a Poisson process with rate λ=1 per minute. The number of copies made is uniformly distributed between 1 ...
2
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1answer
26 views

In a school the odds of a student speaking spanish are $30\%$. [duplicate]

If we select $3$ random students, what are the chances of at least one of them speaking spanish? So, I saw this question and tried to solve it, seemed like an easy question but I was wrong and still ...
1
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1answer
14 views

probability of a positive random variable larger than a sequence tending to 0

Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, P)$ such that $X > 0$. The distribution of $X$ is not known. Let $\{ a_k \}_{k = 1}^\infty$ be a sequence such that $a_k ...
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0answers
24 views

Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the ...
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2answers
20 views

Hazard rate question (Exercise 4.4.7 from Grimmett and Stirzaker)

Exercise 4.4.7 asks for the hazard rate of $X$ where $X$ has the Weibull distribution: $$ P(X > x) = e^{-\alpha x^{\beta - 1}}{\rm \hspace{1cm} where\ } x \geq 0. $$ I computed the answer to be ...
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0answers
46 views

Probability of number of people who know a rumor

Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't ...
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1answer
24 views

How many times do I have to perform event x (with a probability of y) to ensure that it happens z times with a confidence of 95%?

Preferably in a formula please. E.X. How many times do you have to roll a dice to have a 95% chance rolling a total of five sixes?
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1answer
21 views

Is this proof for almost surely convergent valid?

If I want to show that a sequence of random variables $X_n$ has the property $P(X_n\rightarrow 0)=1$, is that enough to show $\forall\epsilon>0,P(|X_n|\ge\epsilon)\rightarrow 0$? I think the ...
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1answer
30 views

What is $E[Z|Z\ge 0]$ where $Z$ is a continuous random variable with support in $[-1,1]$?

I have a random variable $Z$,I seek an expression for $E[Z|Z \geq 0]$. I assume this is easy to get hold of but I just can't seem to get it. As a further complication $Z=X-Y$, where $X$ and $Y$ are ...
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0answers
14 views

Problem with compound Poisson process

Let $X_k$ for $k=1,2,...$ be a sequence of i.i.d. random variables with $\mu_k=0$ and $\sigma_k^2=1$ for all $k$. Consider de random process $$S(t)=\sum\limits_{k=1}^{N(t)}X_k $$ where $N(t)$ is a ...
2
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3answers
64 views

$X,Y$ are iid from distribution $F$, which is a continuous function, then is $P(X=Y)>0$?

Suppose $X,Y$ are iid random variables from a distribution function $F$, which is a continuous function. Then is it always true that $P(X=Y)=0$? For me, the answer is trivially YES. We have $\int_y ...
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0answers
18 views

Difference in real vs computer simulated probability distribution results.

If this is not the right place to ask this,please guide me. I had a thought about what if our basic laws are somehow flawed such that they work in the situations we have observed but not in some ...
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0answers
42 views

Question on the hitting probability

I am confused on the relation between some basic operations with sets and probability. Consider a set $$ A=B\cup C\cup D $$ with $B,C,D$ disjoint sets. Take a random set $S$ almost surely non ...
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0answers
18 views

Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?

Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
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1answer
11 views

Use of Bayes theorem in the Lovásk local lemma

Here's a line from the proof on Wiki I don't understand. $$\Pr(A\mid\bigwedge_{B\in S}\bar{B}) =\frac{\Pr(A\bigwedge_{B\in S_1}\bar{B} \mid \bigwedge_{B\in S_2}\bar{B})}{\Pr(\bigwedge_{B\in ...
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1answer
36 views

what is the CDF of $f(x)=\frac{3x^2}{2}$?

This is probably a dumb question but I just want to make sure. The pdf is $f(x)=\frac{3x^2}{2}$ if $-1 \leq 0 \leq 1$. The CDF is $F(x)=\frac{x^3}{2}$ but with what bounds? sorry if this is an easy ...
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1answer
26 views

Sending bits and parity bits over noisy channel

Consider a sender is trying to send three information bits $a_1$, $a_2$, and $a_3$ over a noisy channel with error probability $$p = 0.001$$ That is with probability $p$ each bit may be flipped ...
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0answers
11 views

Is the mixture of Exponential family distributions an Exponential family distribution too?

Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. ...
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0answers
25 views

Proof $\lim\limits_{n\to\infty} \dfrac{|S_n|}{A_n^{1/2} (\log_2 A_n)^{1/p}(\log_2\log_2 A_n)^{(1+\delta)/p} } = 0$

Let $\{X_k\}$be a random variables sequence and $S_n=\sum_{k=1}^n X_k$. I have $$ \limsup\limits_{n\to\infty} \dfrac{|S_n|}{A_n^{1/2} (\log_2 A_n^2)^{1/p}(\log_2\log_2 A_n^2)^{(1+\delta)/p} } \le ...
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2answers
39 views

How to find the right way to solve a given problem?

We distribute 10 indistinguishable balls to 5 girls. All the distributions have equal probability Let X be the number of girls who get at least 1 ball I need to find $Pr(X=3)$ and ...
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0answers
51 views

The expected value of the smallest number in sample $S$ is:

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
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0answers
25 views

Expected maximum of maxima

Let $F(x)$ denote some CDF, and $\{f_i\}_{i=0}^m$ be a set of random variables independently drawn from that distribution. I would like to compute $$ E\bigg[ \max\bigg\{ \max\bigg\{\{f_i\}_{i=0}^m ...
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2answers
19 views

normal approximation with continuity correction

a fair die is rolled 100 times. What is the probability that "6" appears more than 15 times? Use the normal approximation with continuity correction. I've found the mean to be $100/6$ or $50/3$ and ...
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0answers
28 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
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1answer
22 views

How to set the limit for interated integral of $f(x,y)$ over diagonally partitioned region

I would like to compute $$I = \int_{\mathcal{R}} f(x,y) d\mathcal{R}$$ $$ f(x,y) = \begin{cases} x^2, \quad 0 < x < y < \pi \\ y^2 , \quad 0 < y < x < \pi \end{cases}$$ ...
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1answer
15 views

coincidence of recurrent random processes with infinite expected periods

That subject might not be quite accurate, but let me clarify. At discrete times t=1,2,..., with probability 1 events of type X and Y produced by independent random processes happen infinitely often, ...
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1answer
30 views

How many draws to have a 90% chance of getting the ace of spades?

You have a standard 52-card deck, and you want to take the minimum number of draws from a random/shuffled deck such that you have a 90% chance of drawing the ace of spades. How would you find the ...
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1answer
35 views

Absolute value of a random variable

I have never encountered this concept before. Is this equation valid for $y>0$? $$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$ What about this? $$\mathbb{P}(|X|>y) = ...
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1answer
16 views

Probability problem involving Bayes' Formula

Two brothers share a car. They each have n keys in their pocket. They try one key at random and discard it until they can get the right one to start the car. Brother $A$ has only $1$ compatible key in ...
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1answer
23 views

What is the probability that smallest number is $6$ and largest is $15$?

Five numbers are drawn without replacement from the numbers $\{1, 2, 3, \ldots, 20\}$. What is the probability that the smallest number is $6$ and the largest is $15$? I am studying for a stats ...
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1answer
36 views

A probability problem about a thief?

Imagine that a detective is 60 percent sure that Mr.X is the thief in an investigation. 2 days later, They find new information about the real thief. The real thief is left-handed. Mr.X is left-handed ...
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1answer
39 views

Differences of heads and tails in a fair coin

I'm very new to this so I would appreciate a detailed explanation. I wrote a very simple Matlab program that "flips a coin" (randi([1 2])) $n$ times. Every time I ...
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1answer
24 views

What is the probability Amy wins a lottery prize for correctly choosing 5, not six, numbers…

Here is the full question: What is the probability that Amy wins a lottery prize for correctly choosing 5, not six, numbers out of six integers chosen at random from the integers between 1 and 40 ...
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0answers
18 views

Arrival times of taxis is poisson process

You are waiting in line for a taxi. There are two people ahead of you. Taxis arrive in a Poisson process, at average rate of one every two minutes Let T$_2$=time until 2nd taxi arrives I'm trying to ...