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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Conditional expectation over a convex set

Let $\boldsymbol{X}$ be an $\mathbb{R}^d$-valued absolutely continuous and integrable random vector. Let $L \subset \mathbb{R}^d$ be a closed convex set. Does it hold that $\mathbb{E}[\boldsymbol{X} \ ...
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1answer
9 views

Mean age among employees in a company.

In a company there are 32 men and 59 women. Male mean age is 48.5 and female 39.2. One of the women ended working at the company and was replaced with a 23 year old man. Calculate the employees ...
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15 views

Probability to draw all items (Simplification)

I was calculating the probability to draw all items in a list of N items, by picking one randomly, replacing it in the list, etc. I found this formula after n picking. $$ \sum_{i=0}^N ...
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1answer
22 views

Average distance between two random points in a square

A square with side $a$ is given. What is the average distance between two uniformly-distributed random points inside the square? For more general rectangle case, see here. The proof found there is ...
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1answer
15 views

Confusion about probability question

Some friends are sitting together playing a game that involves rolling dice. On one turn, a player rolls six 6-sided dice and gets one of each number showing. Another player sees this and asks "what ...
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1answer
11 views

How to get the value of 'scaled' binomial distribution?

People kindly told me that there is not a equivalent popular distribution for $aX$ when $X$ is distributed as Binomial, but it is just a 'scaled' distribution. Here, $a$ is a positive constant. ...
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1answer
16 views

Poisson Probability (Shopkeeper Sales)

SOLUTIONS: (A) 0.1804 (B) 0.0166 (C) 0.3233 Mean = 2/7*5 (a) x = 3 (b) x > 5 I'm still unsure how to approach each question, because I still get the wrong answers.
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1answer
24 views

pdf: What is the distribution of aX when X ~ Binomial / Gaussian

Question When $X$ is distributed as binomial or Gaussian, is $aX$ equivalent to some famous distribution? Here, $a$ is a real and positive number. Background I know a general formula giving $aX$'s ...
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1answer
23 views

Joint Probability with many values

Consider I have the following tree structure which provides the relation between various entities. Associated with this, I have the following table with data. ...
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0answers
22 views

Convert min to max probability

Assuming $Y=min(q_1,\ldots,q_n)$ , $q_i\sim N(\mu,\sigma^2)$ I want to express $Y$ in terms of the $Q$ function. Knowing that $Y=min(q_1,\ldots,q_n)=-max(-q_1,\ldots,-q_n)$ $P(Y\leq y)= ...
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0answers
11 views

Comparing results of calculated probability and practical probability [on hold]

I am planning to compare probability that comes from theory and practical experiment. So here the detail of my experiment: I have black box B where there are N lines as input and N lines as output, I ...
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0answers
13 views

Poisson Distribution Worded Problem (Typist & Corrections Question)

The rate is 1/800 The mean is 1/800*(200) (a) 1 - poissCdf(1/800*200,0,1) = 0.026499 0.026499 = Probability that a page is deemed unsatisfactory OR Probability that a page needs to be retyped ...
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10 views

Most efficient estimator

$X_1,...X_n$ is a random sample of size $n$ from a population with mean $\mu$ and variance $\sigma^2$.There are three estimators for $\mu$:  $\hat\mu _1=\frac{x_1+x_2}{2}$ $\hat\mu ...
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0answers
9 views

Is there any simple formula for this probability distribution of random walk?

Assume $\{S_n\}_{n\geq 0}$ transits as follows: $S_0=0$, for $k\geq 1$, $P(S_{n+1}=k+1|S_n=k)=\alpha$, $P(S_{n+1}=k|S_n=k)=\beta$ and $P(S_{n+1}=k-1|S_n=k)=1-\alpha-\beta$, where ...
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22 views

Game of Keno from Sheldon Ross Chapter 4

I am facing with the following problem: A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are selected at random by the casino from the set of numbers 1 ...
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1answer
16 views

Expected value of Bernoulli with probability of success Gaussian distributed

I have a circle with centre $(0,0)$. I am generating Matlab code to include $N$ neurons in a neural network. The probability of including individual neurons in a network decays exponentially with ...
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0answers
30 views

Calculating the probability of being caught

This is a game theory problem I am working on. I apologize if this question is elementary; my probability is pretty rusty and I'm also new to this - I just started working on my PhD and after being a ...
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0answers
28 views

Random Walk Question - what is the probability of eventually reaching the origin? [duplicate]

Consider the random walk $S_n$ given by $S_{n+1} = S_{n} + 2$ with probability $p$ ; $S_{n+1} = S_{n} - 1$ with probability $1-p$. Assume that $S_0 = n > 0$ with certainty. What is the ...
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0answers
22 views

Is this Markov Chain irreducible? Aperiodic? What is its equilibrium mass function? [on hold]

Consider a Markov chain with outcomes $\{0,\dotsc, n\}$ and transition probabilities \begin{align*} P_{i,i+1} &= p \\ P_{i,i-1} &= q \end{align*} for $1\leq i \leq n-1$ and $p+q=1$. Assume ...
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0answers
14 views

Show the equilibrium vector of a transition matrix for a Markov Chain has no zero entries [on hold]

Let P be a transition matrix for a regular Markov chain and let w be its equilibrium vector. Show that w has no zero entries.
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1answer
33 views

What distribution it is based on the histogram? [on hold]

I generated this histogram in r and was trying to determine which distribution I should use, my guess is normal or Binormial. But I'm not sure, can anyone help please?
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1answer
19 views

Setting up the expected value for $x_t=\sin(2\pi U t)$.

We have the series $x_t=\sin(2\pi U t)$ where $t=1,2,3,\ldots$ and $U$ is uniform on the interval $(0,1)$. I have to find the expected value of $x_t$. I always thought that if $X$ is a continuous ...
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1answer
13 views

Joint Density Functions

Let X have density f(x)=2x for 0 a) Give the joint density function of (X,Y). Calculate the probability P(Y-X>1.5) Since X and Y are independent, can I just say that f(x,y) is just equal to 2x? And ...
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1answer
47 views

Is conditional probability $P(A\mid B)$ proportional to $P(B\mid A)$?

It feels a bit odd but since $$P(A\mid B) = \frac{P(A,B)}{\sum_A P(A,B)} \propto P(A,B)\text{ and }P(B\mid A) = \frac{P(A,B)}{\sum_B P(A,B)} \propto P(A,B)$$ can we say that $P(A\mid B) \propto ...
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1answer
13 views

A question about iid observatins $(X_1, \cdots ,X)n)$, knowing that $f_X(x) = ve^-vx$ , with x>0 and v>0.

How do I show that X also have gamma distribution with parameters $nv$ and $n$? I know about the relationship between exp and gamma distributions, but i don't know how to solve this.
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0answers
17 views

Multiple Anihilating Random Walks in a Ring (cycle)

I've been trying to solve this problem for a long time. Problem Let $R$ be a cycle with $2n$ nodes and assume there are $2k$ particles performing a simple random walk in this ring (i.e., they have ...
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0answers
7 views

Calculating variance & expected value of a statistic with exponents

I am trying to calculate of a statistic: $Var(\frac{1}{1 + 1/n \sum_i x_i})$. Thus far, I have $=E[(1 + 1/n \sum_i x_i)^{-2}] - E[(1 + 1/n \sum_i x_i)^{-1}]^2$. How do you deal with exponents inside ...
8
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2answers
830 views

Curious about a made-up paradox

I have thought up a paradox, that may already exist, but I do not know what it's called. It's bothering me though, so any help regarding solving it or proving it impossible would be appreciated. In ...
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0answers
14 views

Calculating normalization constant in circle detection process

I'm doing some research in computer vision, and I need to calculate if two edge points correspond to the same circular object, but i have few questions. Formula is: where: pi and pj are two ...
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1answer
25 views

Definition of an absolutely continuous random variable

Just what is the proper definition of an absolutely continuous random variable? It's supposed to be something like: $$\mathbf{P} (A) = \int_A f d \mu$$ for some Borel set $A$. But what is $\mu$? Is ...
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0answers
18 views

Naive Bayes' classifier [on hold]

Here's the problem set: I got the first two sections down but I have no idea how to do the third section. Can anyone help?
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2answers
35 views

Probability of a pair of red and a pair of white socks among five chosen

In the box are $7$ white socks, $5$ red socks and $3$ black socks. $2$ socks are considered a pair if they have the same color. $5$ arbitrary socks are selected at random from the box. ...
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0answers
26 views

proof that some expected value equal to $\theta (\log n - \log k)$

So here is the problem - Given the following equation: $(c_2\cdot \log n) - (c_1\cdot \log k)\le E(X)\le 1+ (c_1\cdot \log n) - (c_2\cdot \log k)$ When $c_2,c_1\gt0$ and also $c_1\gt c_2$ In ...
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1answer
27 views

Geometric Brownian Motion [on hold]

I am new there. How can I calculate following expected value: $$E[X(s)\times X(t)]$$ where $X$ is Geometric Brownian Motion, i.e. $X(t) = exp[(\mu - 0.5\cdot \sigma^2)t + \sigma\cdot W(t)]$ ...
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0answers
23 views

Probability theory's problem [on hold]

We number a regular icosahedron's faces (it has 20 faces) and start to throwing up randomly, and note the number of the face which it has arrived. Writing down the numbers until the sequence of the ...
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2answers
35 views

Probability to get from point A to point B.

In the photo each dot is a city and each blue segment a road. Each road is blocked with probability 1/3 and free with probability 2/3 (independence among all roads). What is the probability that it is ...
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0answers
15 views

How to show the series of expectations for truncated symmetric random variables is convergent

Suppose that $(X_n)$ is i.i.d. with symmetric distribution and that $E(|X_1|)<\infty$. I want to show that $\sum\limits_{i=1}^{\infty} \frac1iE(X_i 1_{[|X_i|<i]}) $ converges. Attempt: Since ...
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1answer
16 views

Sampling distribution question with unknown n.

Suppose that 53% of the population of voters were in favor of fighting the global warming. If we wanted to conduct a random sample of size $n$ of voters, how many should I survey if I want the ...
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0answers
55 views

Random Walk Without Repetitions

Suppose that we simulated a random walk on $\mathbb Z$ starting at $0$. At each step, we transition from position $x$ to position $x-3,\,x-2,\,x-1,\,x+1,\,x+2,$ or $x+3$ with equal probability. If ...
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1answer
14 views

Rescaling a probability

I can't ge me head around this. I know that between 00:00h and 00:30h (i.e. within 30 minutes) a person is with a chance of 90% in room A, 7% in room B and 3% in room C. Now the task is, to derive a ...
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0answers
17 views

Mean and variance question

An electronic device periodically records the voltage applied at its input, truncating it to the nearest integer in each case. Under the usual assumptions, evaluate the mean and variance of the error ...
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3answers
27 views

Finding the probability of an event with binomial distribution using a normal approximation

A Tarheels basketball player is obsessed about practicing his free throws. It is known that he is $75\%$ free throw shooter. One morning he decides to shoot $100$ free throws. You may assume that ...
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1answer
31 views

probability over 3 values with dependency

At the exercise, there is no information that B and C are independent, but with logical reasoning, there must be a pendency. The problem is, I can not create a connection with depency of B and C, is ...
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0answers
12 views

UMVUE for altered Normal distribution

Let $X_1 , ...,X_n$ be a sample from a normal population $N(\mu , \sigma^2)$. It's easy to find the UMVUE for $\mu$ and $\sigma^2$: (1) After finding the joint density of X=$(X_1 ,...,X_n)$, we find ...
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0answers
22 views

Analysis of sorting Algorithm with probably wrong comparator?

It is an interesting question from an Interview, I failed it. An array has n different elements [A1 .. A2 .... An](random order). We have a comparator C, but it has a probability p to return correct ...
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0answers
15 views

Identifying markov chains and the markov property [on hold]

Im currently revising for a probability exam and I came across this question: Let $(X_n),n\geq1$ be a sequence of independent identically distributed non- negative random variables taking values in ...
2
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0answers
26 views

How to show that $p(t|x,\mathbf x,\mathbf t)= \int p(t|x,\mathbf w)p(\mathbf w|\mathbf x, \mathbf t)d\mathbf w $

The following paragraph is approximately cited from Bishop's book, Pattern Recognition and Machine Learning. In curve fitting problem, we have training data $\mathbf x$ and $\mathbf t$, along ...
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0answers
18 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
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0answers
32 views

I need help to solve this complex question [duplicate]

Peter has 12 pairs of socks and 6 pairs of gloves in different colors. His socks are in green, yellow, black, and grey (3 pairs each). Peter's gloves are either blue, black, or red (2 pairs each). ...
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2answers
39 views

Tricky Cardinality Question/Riddle [on hold]

Mike is a child and he is playing with lego box. He begins by counting the number of pieces in his lego box. He recognize that some pieces are green, some pieces are blue, and some are green and blue. ...