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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5 views

Conditional law given a sigma algebra

Let $f : x \longmapsto $min{$x,1-x$} and ($U_n$; n $\geq 1$) an IID sequence with uniform law on $[-1,1]$. For $x \in [0,1]$ we define the sequence $(X_n)_{n \geq 0}$ with $X_0 = x$ and $X_{n+1} = X_n ...
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0answers
14 views

Understanding of the probability

I have a problem with understanding some of my statistics homework. I hope that some of you could help me understand. In summary the question is as follows: There are 30 people in a group, which ...
0
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0answers
17 views

Vacuous Probabilities

Suppose it we have that for all $x \in X$, $\neg ( Ax \wedge Bx)$ holds. Then can we say anything about the value of $P(Ax \mid Bx \wedge x \in X)$? Can we say, for example, any of the following? ...
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1answer
25 views

Integral involving CDF of a normal distribution

Can we evaluate the following integral ? $$\int_0^\infty x e^{-x^2} \Phi(ax+b)\,\mathrm dx$$ Here $\Phi(\cdot)$ is the cumulative probability distribution function of a standard normal random ...
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0answers
21 views

Independent random variables and integrability

This is a problem that I am stuck at. I think I have to prove the hint first. But I can't find a way to prove the 'only if' part of the hint. (the 'if' part is just manifest). Could anyone help me ...
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0answers
14 views

A minimum settlement for a bargaining problem

Question: Alpha and Beta are 2 companies. Now Alpha thinks that Beta has violated Alpha's trademark. Beta denies that. Now, Alpha is threatening to go to the court and claim 5,000,000 EUR from Beta ...
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0answers
13 views

Distribution of the sum of random variables

Let $X_{1}$,$X_{2}$,...,$X_{N}$ be a Dirac distributed (not independent) random variables. What is the distribution of $\sum_{i=1}^{N}{X_{i}}$?
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2answers
28 views

Given the probability distribution of X, whats the PDF of X²?

Let's say we have a random variable $X$ with a certain probability density function $f_x(x)$. 1) How should I find out the PDF of the random variable $X^2$? 2) Any ideas for the PDF of $T = ...
1
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2answers
104 views

Convergence in Probability

Consider a sequence of $N$ Bernoulli trials with, with probability of success denoted by $p$, and let $X$ be the number of successes. Show that as $N\rightarrow\infty$, $\frac{X}{N}$ converges in ...
2
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2answers
38 views

How many stones are white?

Assuming there's a pipe infinitely generating black or white stones randomly, with each color having 50% chance. Now someone randomly get 105 stones from the pipe, then randomly put 100 in bowl A and ...
0
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1answer
27 views

Probability of 2 cards being the same suit playing 3 different hands

I understand how to do this for a single hand. 12/51 which is about 24% But what if I am playing three separate hands. What is my chance of one of those hands having two suited cards? I'm not sure ...
0
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1answer
15 views

Probability function of Acos(x)

Let's say I have a signal $y(t) = Acos(2\pi f_c t)$, where $f_c$ is the carrier frequency and $t$ is the independent variable. Since I work with discrete signals i sample this signal with a sampling ...
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0answers
9 views

Expected number of attempts to completely cover a set (with repetitions)

I have a bag with $n$ balls with numbers from $1$ to $n$ in it. I pick a ball, note its number and put it back into the bag. I then keep repeating this step until I have picked up all $n$ balls at ...
2
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1answer
21 views

Do absolutely continuous random variables have a continuous distribution function

If we define absolutely continuous random variables by Lebesgue integrals & Lebuesgue measures, i.e. $$F(t) = \int_{-\infty}^{t} f(x) d x$$ for some Lebesgue integrable $f(x)\ge 0$, is it always ...
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0answers
12 views

The integration of a Gaussian process.

Now I'm reading this post: Distribution of integral of a normally distributed random variable Suppose $r(t),t\in[0,T]$ is a Gaussian process.I want to show that $$\int_0^Tr(t)\,dt$$ has normal ...
2
votes
1answer
28 views

To show $X$ and $|X|$ are not jointly continuous

Suppose $X\in N(0,1)$. Show that $X$ and $|X|$ are not jointly continuous. I am not sure how I can approach this problem. But the following method seems plausible to me: $$P(X\leq ...
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0answers
29 views

Birthday problem with a twist

Two of my facebook friends had their birthdays on the same day The first guy's name was "Wael Toujeni" The second guy's name was "Wael Jeni" How do I calculate the probability of this event ...
0
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1answer
16 views

conditional distribution for a discrete RV

For a discrete RV $X$, is it true that the conditional distribution $P_{X \mid Y} (B \mid y)$ is discrete as well for all $y$? I only managed to prove that this is true almost surely. Let $\Pr(X\in ...
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1answer
42 views

Probability question for economics that I'm struggling with. Please help.

(There are 4 districts in the land of Oz. At home, the inhabitants of each region wear ties of a special colour, Munchkins (M) wear blue, Scarecrows (S) wear purple, Tin Men (T) wear red and Wizards ...
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2answers
89 views

Boy and girl paradox is driving me crazy

I know this question is asked over and over, but I still can't understand anything. Say I'm introduced to a random father of two and I want to know what's the probability that both his children are ...
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0answers
13 views

Probablity with percentage [on hold]

An amplifier is made up of three transistors, $A, B$ and $C$. The probabilities of $A, B$ or $C$ being defective are $\frac{1}{20}$,$\frac{1}{25}$ and $\frac{1}{50}$ , respectively. Calculate the ...
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0answers
17 views

A Quick Question on Filtrations and Stopping Times

Let $(\Omega,\mathcal{F},P)$ be a probability space with filtration $\{\mathcal{F}_t:t\geq0\}$. Define $\mathcal{F}_{t+}=\bigcap\limits_{s>t}\mathcal{F}_s$. If $\tau$ is a random time and $\forall ...
0
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2answers
22 views

Finding Probability of different Dice Rolls

A fair die is thrown three times: What is the probability of getting: three sixes? What is the probability of getting: six, one, six? My solution: Probability of getting three ...
1
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1answer
18 views

Find number of circular arrangements possible

If 20 persons were invited for a party, in how many ways will two particular persons be seated on either side of the host in a circular arrangement? According to me the answer should be $17!.2!$. But ...
0
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1answer
21 views

Find Combined Probability of one Coin and at most three Dice

A fair coin is tossed: If heads: an unbiased die is thrown three times. The sum of the outcomes of the three rolls is recorded. If tails: an unbiased dice is thrown once. The result is ...
3
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1answer
27 views

Almost sure convergence and L1 convergence

I am preparing myself for the mid-term exam of my probability theory exam, and am solving questions from previous years exams. One of these questions I couldn't answer, and so far I haven't found ...
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0answers
11 views

find a sequence of independent, non-identical r.vs , such that $limsup (X_1 + … + X_n) / n = \infty $ [on hold]

Find a sequence of independent, non-identical, non-negative random variables with $E X_i = 1$, such that, $$limsup_{n\rightarrow \infty} \dfrac{(X_1 + \dots +X_n)}{n} = \infty \ a.s.$$
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1answer
18 views

A question on joint probability density functions.

I know that the pdf $X$ conditional on $Y=y$ is $$f_{X|Y}(x|y)=\frac{f_{(X,Y)}(x,y)}{f_Y(y)},$$ and this can be used to calculate conditional probabilities such as $P(X>\alpha | Y>\beta)$ (for ...
0
votes
1answer
23 views

Find Combined Probability of one Die and two Coins Tossed

A fair die and two unbiased coins are tossed. What are possible outcomes of each object and the probability of each outcome? My solution: Probability for a fair Die $D$: $\frac{1}{6}$ ...
0
votes
1answer
27 views

Probability: Understanding Random Variables

I am using Ross' A First Course In Probability (4th). On page 113, Example 1d states the following: Independent tirals consisting of the flipping of a coin having probability $p$ of coming up ...
0
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1answer
12 views

Transformation of Random Variables Involving Division

I have been working on this problem from a previous exam in Probability theory but I can't understand the next step I am supposed to take. Here is the problem: Suppose that $Z_1$ and $Z_2$ are ...
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0answers
19 views

Probability of winning three prizes in a raffle [on hold]

What is the probability/odds of winning 3 raffle prizes with just 3 tickets and 10 prizes? (90 entrants, all have 3 tickets, he won them at random times, not all at the start.)
4
votes
1answer
71 views

Is there any short proof of this classical problem?

Let $X,Y$ be two i.i.d. r.v.'s with zero mean and unit variance. If $X+Y$ and $X-Y$ are independent, then $X$ and $Y$ are both standard normal distributed. Is there any short proof for this problem?
4
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0answers
25 views

shifted exponential distribution with inter-arrival time

Given that time interval $T^*$ in seconds between certain events has a negative exponential distribution. The instrument cannot detect intervals which are less than $\delta$ seconds. Let $T_1, ..., ...
13
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4answers
2k views

A coin is flipped ten times. What is the probability that the first three are heads if an equal number of heads and tails are flipped?

I understand the question but I am not sure how to solve it. For example, if we flip HHHTTTTT then the next three must be heads because of the question. This however seems counterintuitive. I believe ...
1
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1answer
30 views

probability about animals being moved

I have this scenario: 1 animal with 30% probability of be moved to Japan. 1 animal with 30% probability of be moved to Japan. 1 animal with 30% probability of be moved to Japan. 1 animal ...
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0answers
12 views

Evaluating an expected value in Jeffrey's prior for binomial distribution

The material I'm reading derives Jeffrey's prior (or rather, the Fisher information for the Jeffrey's) for single-parameter binomial distribution in a manner quite similar to this Wikipedia article. ...
2
votes
1answer
34 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even tough its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
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2answers
22 views

Questions about joint probability, conditional probability, and Bayes' rule and their relationships [on hold]

Joint probability: $P(X|Y) = P(X,Y)/P(Y)$ Conditional probability: $P(X|Y) = P(Y|X) * P(X)/P(Y)$ My questions: How do you know when to use the above joint probability equation as opposed to just ...
0
votes
1answer
13 views

How many different sets of 8 elements can I pick if I am picking from a bag of 1681 elements probability and counting [on hold]

I have 1681 points and trying to see how many different constellation of 8 points I can have to see if it is feasible to try out all possibilities to find the best. It's actually a Communication ...
2
votes
2answers
30 views

What is the probability of two dice getting a sum of 7 without a two?

I am currently working on conditional probability and I am somewhat confused about how exactly to complete this problem. I know that to find conditional probability that you utilize: $$P(A|B) = ...
0
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0answers
12 views

Product of normal densities in a Bayesian context

Two analysts, analyst A and analyst B, are interested in the probability distribution for a multivariate-normal vector $X$ with five dimensions. A estimates a density function $f_X(X=x)$ for $X$, ...
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votes
1answer
27 views

Simple dice probability question [on hold]

Just a simple quick question. Five dice are thrown at the same time, all results are independent of each other. The dice are regular dice, with 6 sides. What is the probability that exactly three ...
1
vote
2answers
25 views

Let $A,B,C$ be events. Find an expression for the event “at least one of B and C occur, but A does not”

Let $A,B,C$ be events. The event "$A$ and $B$ occur but $C$ does not" may be expressed as $A \cap B \cap C^c$. (a) Find an expression for the event "at least one of B and C occur, but A does not" ...
0
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0answers
20 views

Cartesian product of two markov chains is a markov chain [on hold]

Is a Cartesian product of two Markov chains a Markov chain? And is it true for a countable product as well?
0
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1answer
26 views

Tossing a coin with two random variables, conditional probability.

I have a question regarding conditional probabilities. Experiment: we toss a coin $10$ times. We count the amount of head and we toss that amount again. Let $X$ be the amount of heads in the first ...
0
votes
1answer
23 views

Probability of two RVs being equal

Let $X$~Binom($n,1/2$) and $Y$~Binom($m,1/2$) be independent. Calculate $P(X=Y)$. My attempt: Assume $m\le n$ $$P(X=Y)=\sum_{k=0,\ldots,m} P(X=k)P(Y=k)=(\frac{1}{2})^{n+m} \sum_{k=0,\ldots,m}{n ...
2
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0answers
40 views
+50

Equivalence between conditions for convergence

Let $(X_k)$ be independant random variables such that $X_k\sim\mathcal{P}(p_k)$ (Poisson distribution with parameter $p_k$). So in particular we have $ \sum_{n=1}^NX_k \sim \mathcal{P}(\sum ...
2
votes
1answer
21 views

Random number generator from a piecewise PDF

I'm trying to create a random number generator on the interval $(a,c)$ given a probability density function defined as: $$f(x) = \left\{ \begin{array}{lr} \dfrac{C}{x} &, x \in (a,b)\\ ...
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2answers
4k views

Find the probability density function of $Z = \max (X, Y ) \mbox{ and } Z=X+Y$

Let $X$ and $Y$ be random variables that are uniformly distributed between $0$ and $1$. a) Find the PDF of $Z = \max (X, Y )$. b) Find the PDF of $Z = X + Y$ .