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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2answers
22 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 ...
0
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4answers
46 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 ...
4
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2answers
21 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 ...
4
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0answers
45 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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3answers
37 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 ...
1
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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
9 views

Probability Help with finding mean and variance of estimators

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:$$ ...
0
votes
1answer
15 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 ...
0
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0answers
14 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 ...
0
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1answer
19 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 ...
1
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1answer
13 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 ...
0
votes
1answer
16 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 ...
2
votes
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 ...
0
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0answers
15 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 ...
17
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6answers
63k views

A family has three children. What is the probability that at least one of them is a boy?

According to me there are $4$ possible outcomes: $$GGG \ \ BBB \ \ BGG \ \ BBG $$ Out of these four outcomes, $3$ are favorable. So the probability should be $\frac{3}{4}$. But should you take ...
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 ...
0
votes
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 ...
0
votes
0answers
10 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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2answers
428 views

How many ways can you deal a hand of 13 playing cards that contains four spades, seven diamonds, and two hearts?

How many ways can you deal a hand of 13 playing cards that contains four spades, seven diamonds, and two hearts? leave the factory form is fine thanks!!!
1
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0answers
23 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 ...
0
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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 ...
3
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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 ...
2
votes
0answers
27 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 ...
2
votes
0answers
302 views

Finding joint distribution function?

Let $U$ and $V$ be two independent uniform $(0,1)$ random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the ...
4
votes
2answers
58 views

Probability of winning dice roll-off with a re-roll

I am looking to find the probability of winning a basic dice roll-off using a 6 sided die if one of the players can re-roll their die. The main thing that is throwing me off is that player 2 can ...
-1
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0answers
28 views

What is the limit distribution of this sequence of random variables? [closed]

Find the distribution of the limit when $n\to\infty$ of$${S_n\over n^2}$$ where $S_n=X_1+X_2+...+X_n$, and $ X_1,X_2,...$ are random variables i.i.d. with characteristic function ...
1
vote
2answers
48 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 ...
0
votes
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 ...
1
vote
1answer
34 views

Order Statistics with two Groups of Draws

Let $X_{1},X_{2},\ldots,X_{m},\ldots,X_{n}$ be independently drawn from a distribution $F$ and let $Y_{k}^{(n)}$ be the $k$-th order statistic (Convention: ...
0
votes
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?
1
vote
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 ...
1
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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 ...
0
votes
1answer
25 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 ...
0
votes
0answers
13 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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.
2
votes
2answers
385 views

Finding covariance from marginal densities.

A quarter is bent so that the probabilities of heads and tails are 0.40 and 0.60. If it is tossed twice, what is the covariance of Z, the number of heads obtained on the first toss, and W, the total ...
0
votes
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. ...
-1
votes
2answers
49 views

What is right solution for this probability problem?

This drug can cure $90$% of all diseases. What is probabilty of successful healing at least $18$ people of $20$ people, who have taken the drug? What is the right solution and why? From my point of ...
-1
votes
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 ...
0
votes
1answer
2k views

Joint cdf and pdf of the max and min of independent exponential RVs

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
172 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
1
vote
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, ...
0
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0answers
39 views

For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
1
vote
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 ...