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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0
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1answer
17 views

Let $X$ and $Y$ be square integrable random variables s.t. $E[X|Y]=Y$ and $E[Y|X]=X$. Prove $P(X=Y)=1$.

Let $X$ and $Y$ be square integrable random variables s.t. $E[X|Y]=Y$ and $E[Y|X]=X$. Prove $P(X=Y)=1$. Furthermore, when the condition changes to $X$ and $Y$ are integrable, show that the conclusion ...
0
votes
1answer
21 views

The G/M/1 Queue

Please help me with the following. For a long time I am trying to understand the part od the example, which I have attached here The part I don't understand I marked by the red line. It has no sense ...
0
votes
1answer
47 views

What is the probability that Alice and Bob live in the same city?

Problem There are $d$ different cities. Last year Alice and Bob did NOT live in the same city. The probability that Alice moved to a new city since last year is $m$. This probability is the same for ...
0
votes
2answers
86 views
+50

Two length 3 straights vs. one length 5 straight. Which is more likely and by how much?

Using a well shuffled standard $52$ card deck, $2$ players (call them A and B) decide to play a game. They draw community (shared) cards (without replacement) until a winner for that hand is ...
3
votes
1answer
60 views

Walking to infinity stepping on randomly selected lattice points

Suppose you randomly fill the infinite non-negative quadrant of $\mathbb{Z}^2$ with $1$'s and $0$'s, with $1$ occurring with probability $p$ (and $0$ with probability $1-p$). The lowerleft corner of ...
0
votes
1answer
21 views

Let $X$ be normal with zero mean and variance $\sigma^2$, let $Y$ be uniform on $(0,\pi)$. Find the density of $Z=X+a \cos(Y)$.

Let $X$ be normal with zero mean and variance $\sigma^2$, let $Y$ be uniform on $(0,\pi)$ and let $a$ be a real number. Assume $X$ and $Y$ are independent . Find the density of $Z=X+a \cos(Y)$. I ...
-1
votes
2answers
43 views

Coin Flipping - Probability and Value Proposition

Rusty with probability here... The question is: Flip a coin 11 times. If you get 8 tails or less, I will pay you \$1. Otherwise, you pay me \$7. Step 1. Find the expected value of the ...
1
vote
2answers
61 views

Calculate the probability that the running total is exactly n. (homework help)

I am working through Harvard's public Stat 110 (probability) course. Question: A fair die is rolled repeatedly, and a running total is kept (which is, at each time, the total of all the rolls up ...
5
votes
7answers
639 views

Is this lot drawing fair?

Sorry for a stupid question, but it is bugging me a lot. Let's say there are $30$ classmates in my class and one of us has to clean the classroom. No one wants to do that. So we decided to draw a ...
1
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0answers
14 views

Distribution determined by its cgf

It is well known, that if the domain of the mgf $M:=E[e^{uX}]$ of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. Consider the ...
0
votes
4answers
593 views

Parking lot probability

A parking lot on campus has the probability of parking in an illegal spot and getting a ticket is .13, while the probability of finding no park and having to park illegal is .2 In a rushing to class ...
0
votes
1answer
21 views

Probabilistic exponential growth model

I have a real valued number $y_t$. At each time step t, $y_t$ is multiplied by $(1 + \epsilon)$ with probability $p$ and multiplied by $(1 - \epsilon)$ with probability $1 - p$. What is the expected ...
0
votes
1answer
28 views

YACCP: Coupon collector: Pull 20 coupons per time

Yet-another-coupon-collector's-problem: I know this may be a very similar question to others, but I couldn't crack it and this one has a 'special knack' to it, please bear with me: 260 specific ...
-1
votes
0answers
8 views

Markov chain with pair arriving and departure

Does any one know how to find the stationnary distribution of the Markov Chain presented in figure 1. Actually I want to know the probability to be in state 0 (probability that the queue is empty pi(0)...
0
votes
0answers
9 views

Probability distributions

Let $X$ be a Chi-square random variable with $p$ degrees-of-freedom. Let $Y$ be a real Gaussian random variable with 0 mean and variance $\sigma^2$. $X$ and $Y$ are independent. Can we know what the ...
-1
votes
1answer
16 views

How to compute the probability of A union B when B is a subset of A?

GRE - Probability Question Regarding the case in the above problem where B is a subset of A: Intuitively I see that the $P(A \cup B)= P(A)$ in this case. However when I compute the$ P(A \cup B)$ I ...
0
votes
2answers
38 views

Compare two coin tossing games

Compare the following two games: You have a fair coin. After one toss, you will get 1 dollar if you get a head, and 0 dollars if you get a tail. How much will you be willing to pay to play this game ...
1
vote
1answer
383 views

exactly k consecutive heads, n tosses

What is the expected number of strings of exactly k consecutive heads if a fair coin is tossed n times? My current answer is $$ {n-1\choose k} (\frac{1}{2})^{(k-1)} $$ Is this correct? A possible ...
1
vote
1answer
24 views

Simple Logistic Regression - how do I use real data?

Binomial Logistic Regression to predict probability Confusion Point 1: I think I'm right in saying one of the steps of Logistic Regression is to get: $$\log(\mathrm{Odds})$$ Now take this very ...
-4
votes
0answers
25 views

In how many ways can weak law of large number can be proved? [on hold]

Can anyone tell me in how many ways can weak law of large numbers can be proved?(Except using Chebyshev's theorem )
6
votes
1answer
525 views

Different Perspectives of Multinomial Theorem & Partitions

There are 2 important interpretations of the multinomial theorem and coefficients. 1: Determining the number of ordered strings that can be formed using a set of letters. For example, with 1 m, 4 i'...
0
votes
1answer
25 views

Ways of drawing out objects of different types

There are 5 types of objects, each type has many and has the chance of 1/5 being drawn out. I want to find the probability that I can draw out all types in 5 draws. Is it $1*4/5*3/5*2/5*1/5=5!/5^5$ ...
2
votes
4answers
77 views

Interesting probability question - husband and wife committee variation

Twenty husbands and wives (ten couples) are randomly divided into two groups. What is the probability that at exactly 4 wives are in the same group as their husbands? Attempt: There are $\binom{40}{2}...
3
votes
1answer
48 views
+50

Understanding the Skorohod-space

I am having a lack of understanding the Skorhodspace considering cadlag processes. A random variable $X$ is measurable mapping between two measure spaces say $(\Omega,\mathcal{F})\mapsto (\tilde{\...
0
votes
2answers
80 views

Are $X$ and $Y-E(Y|X) $ independent?

The question is quite simple: if $X$ and $Y$ are two random variables (with unknown distribution, possibly independent), are $X$ and $Z=Y-E(Y|X)$ independent? Thinking about the conditional ...
3
votes
1answer
67 views
+50

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
0
votes
1answer
25 views

Comparing variance for two games

I saw this question here Compare two coin tossing games and wanted to figure it out myself, and I had some trouble with variance. I can see easily that both games have expected value $\$500$. But my ...
0
votes
0answers
17 views

Sampling conditional random variables

I am going to generate a sample of random variables conditioning on a linear constraint. To make it clear, suppose that I want to generate multivariate gaussian $(0,\Sigma)$ conditioning on the plane ...
0
votes
2answers
20 views

3 card monte carlo variation

A friend wants to play a betting game with you. There are 3 upside-down cards on the table 2 black and 1 red. Your job is to find the red card. For every dollar you bet he will give you 2 to 1 odds (i....
0
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0answers
42 views

Probability (boxes)

A contest consists in choosing one of three boxes that are covered, inside of which there are envelopes and only one of these envelopes contains the prize. Box 1 contains 8 envelopes, box 2 contains 5 ...
1
vote
2answers
602 views

Statistics question Conditional Probability

Question: Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a ...
2
votes
0answers
27 views

Infinitesimal generator of Brownian motion with additional jumps

A compound Poisson process is a jump process with two parameters, the rate of the jumps $\lambda$ and the distribution of the jumps $\mu$ ($\mu$ is a probability measure on $\mathbb{R}$). The ...
0
votes
0answers
9 views

Computationally Efficient Way to Partition N-Dimensional Space Around Distinct Values

Sorry if the title isn't super helpful, I'm really just looking for someone to point me in the right direction or let me know if there is a standard way of doing this. What I am wondering is, if I ...
0
votes
0answers
52 views

probabillity question [on hold]

Lets say we have 3 balls {1,2,3}, each one is selected by probabillity {p1,p2,p3} I want to calculate the probabillity to choose subset lets say {1,2}, but we need to choose the subset at the same ...
2
votes
4answers
4k views

Addition of two Binomial Distribution

What is the distribution of the variable $X$ given $$X=Y+Z$$, where $Y$~Binomial($n$, $P_Y$) and $Z$~Binomial($n$, $P_Z$)? For the special case, when $P_Y = P_Z = P$, I think that X~Binomial($2n$, $...
2
votes
2answers
26 views

Probability to win

The fine print on an instant lottery ticket claims that one in nine tickets win a prize. What is the probability that you win at least twice if you purchase ten tickets?
2
votes
2answers
220 views

Rolling a die until obtaining the face 6. Whats the expected amout of the sum?

A game where you roll a fair die, repeatedly, adding up the faces that show up, until the face 6 appears. What is the expected sum (including the 6)? All I can think of is that the expected value of ...
0
votes
0answers
21 views

Formula for particular combinations for array of N binary vectors [on hold]

I have arrays with N number of binary vectors and all possible combinations of these vectors, for example for N=3: ...
1
vote
1answer
22 views

Kolmogorov backwards equation / stationary distribution

One can in the case of the Fokker-Planck / forward Kolmogorov equation, set the time derivative term to zero, and solve the remaining ODE to obtain the "forward-time" stationary distribution. Does ...
1
vote
0answers
22 views

Conditional mutual information for continuous random variables

Cover and Thomas provides definition of Conditional Mutual Information (CMI) for discrete random variables but doesn't say anything about continuous variables. Wikipedia has a section about a "more ...
3
votes
3answers
63 views

Flip $n$ coins, discard tails, and continue until $k$ heads remain

Consider the following game: $n$ participants have a fair coin each, on a given round, the not already discarded participants flip their coins, those who flip a tail are discarded from the game, the ...
2
votes
2answers
154 views

Poker probability conundrum

You deal your friend five cards from a standard shuffled deck. He looks at his hand and says either "Oh! I have at least one $X$!" or "I don't have any $X$s," where $X$ is the name of a rank. Your ...
0
votes
3answers
33 views

What is the probability the XOR of binary strings match?

Let $a_1,\dots,a_n$ and $b_1,\dots,b_n$ be independent random binary strings of length $k$. Let $A$ be the bitwise exclusive OR (that is the XOR) of all the binary strings $a_i$ and let $B$ be the ...
0
votes
0answers
24 views

Probabilities in Queues (Exponential and Poisson) [on hold]

I'm having troubles with the next problem: You have 3 servers, servers 1 and 2 serve the people that bought more than 15 items while server 3 serves the people that bought 15 or less. People ...
0
votes
0answers
15 views

Show that a permutation $\pi(t,\omega)$ of a ordering relation is $\mathcal{F}_t$

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration, with $\tau < \infty$. The following definition is ...
2
votes
0answers
31 views

Reflection principle for walk possible steps right, left and stay

I need to use reflection principle for one dimensional walk with equally possible steps right, left and stay. I would like to know if hold a similar identity to that of question Is there an intuitive ...
3
votes
1answer
46 views

Suggestions for Constructing a Random Variables from Correlated Observations

Let $\mathcal{X} \neq \phi $ be a finite set. Let $P_{XY_1Y_2}$ be a fixed joint distribution over $\mathcal{X}\times\mathcal{X}\times\mathcal{X}\ $ and that a random sample $(X,Y_1,Y_2 )$ is drawn ...
1
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0answers
33 views

How to randomly generate two integer matrices $A$ and $B$, so that entries of 3 metrics $A$, $B$, and $AB$ are within certain range?

I ran into this question when writing a program. I need to generate two matrices, and calculate their product. However, I must ensure all entries are within 8-bit signed integer range, i.e. $[-128, ...
1
vote
1answer
20 views

If Mutual Information measures dependence, why is it symmetric?

From Wikipedia we can read: In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. ...
0
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0answers
13 views

Ergodicity in hidden Markov models

Assume that we have a hidden Markov model, where we have a sequence of hidden variables $Z_1, \dots, Z_m$ which form a Markov chain. Now, at each "time point" $i$, an observation $Y_i$ is drawn from ...