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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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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0answers
10 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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13 views

Naive Bayes' classifier

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
24 views

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

In the box are seven white, five red and three black socks. Socks are considered to be a pair if they have the same color. Five arbitrary socks are selected at random from the box. Find the ...
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0answers
18 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
20 views

Geometric Brownian Motion

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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1answer
9 views

Defective Merchandise Probability

If a company currently has a 2% defective rate when making glass bottles, what is the probability that in every 10 glass bottle case, there would be no more than 1 defective bottle? I figured that if ...
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0answers
19 views

Probability theory's problem

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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1answer
23 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
12 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
12 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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39 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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0answers
11 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 ...
1
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1answer
28 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
9 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
18 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
19 views

Probability - winning results 5 games vs 100 games

Game involves rolling two dice: if their sum is 10 or higher, then you win. If it's less than 10, then you lose. What would be your probability of winning at least twice if you played the game 5 ...
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0answers
12 views

Identifying markov chains and the markov property

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
25 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
17 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
31 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
38 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. ...
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0answers
9 views

Under what assumptions is the following first moment monotone?

I'm working on an economic model and am encountering the following mathematical issue. Let $x\sim \mathcal{N}(\mu,1)$, and define $$V(\mu)=\int_0^{\hat x(\mu)}x ...
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2answers
39 views

Can anybody help me to solve this counting problem [on hold]

A color on screens is the result of a combination of three colors red (r), blue (b), and green (g). A color c can be represented by the formula $$c = p_rr + p_bb + p_gg$$ where $$0 ≤ p_r, p_b, p_g ...
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1answer
49 views

Compute Var(x=X1+X2+…+Xn)

Compute $Var(X_1+X_2+...+X_n)$ given $X_1,X_2...$ are iid.,$EX=\mu,Var(X)=\sigma ^2$,and $Var(N)=\sigma [n]^2$, N is a random variable of nonnegative integers independent with X, and my solution ...
1
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1answer
37 views

Probability of asymmetric random walk returning to the origin

Consider the random walk $S_n$ given by $ S_{n+1} = \left\{ \begin{array}{lr} S_n+2 & with & probability & p\\ S_n - 1 & with & probability & 1-p \end{array} ...
2
votes
1answer
19 views

Uniformly Distributed random varibles

Question:Suppose $X$ is a uniformly distributed random variable with possible values $1,2, \ldots, 10$. Compute the expected value and variance of $X$. I have started with making a column ($x$ on the ...
-4
votes
1answer
17 views

Find the 95% confidence interval and interpret the results [on hold]

A random sample of 38 200-meter swims has a mean of 3.96 minutes and the population standard deviation is 0.06 minutes. Construct a 95% confidence interval for the population mean time. Interpret the ...
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0answers
14 views

Using an “auxiliary random experiment” to achieve a desired significance level

My question is somewhat simple, but, nonetheless I am not entirely convinced I am solving it correctly. I need to use the use the Neyman Pearson Lemma to test for $H_o : \theta = .5$ vs. $H_1 : ...
1
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1answer
19 views

Expected value for sum of iid normal variables squared

Let $X_i$ be iid from a $N(\alpha, \alpha)$ distribution. I am trying to find $E[\sum_1^n X_i ^2]$ and thought that I would be able to transform the statistic $\sum_1^n X_i ^2$ into a chi-squared ...
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0answers
23 views

Joint Probability Question

I have a question regarding join distributions. For this question, I have to find the probability that P(X+Y=0). I've attempted multiple different ways to solve this problem, but I keep getting 0 as ...
1
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1answer
17 views

Covariance and Correlation in Multinormal random variable

Find the covariance and correlation of $N_i$ and $N_j$, where $N_1, N_2, \ldots,N_r$ are multinormal random variable. At the beginning, I think that I have: ...
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0answers
8 views

Lists of common sufficient statistics

Can someone suggest a source for common sufficient statistics for exponential families? For example, I'm looking for something more comprehensive than the Wikipedia page for sufficient statistics, ...
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0answers
10 views

Rationale behind formula relating death probability and observed mortality rate

With $M_{x,t}$ stands for the time-$t$ observed mortality rate between ages $x$ and $x+1$ (formulas given below) and $q_{x,t}$ the time-$t$ death probability between ages $x$ and $x+1$ (the ...
0
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1answer
32 views

Mean return time in Markov chain

Given the following Markov chain: $p_{0,1}=1$ (if we are in state 0, we must go to state 1) $p_{i,i+1}=p_{i,i-1}=0.5$ There are infinitely (countably) many states. I assume that $X_0=0$ and define ...
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0answers
36 views

How calculate the probability that there is a row in which there is no silver coin?

There are $n ^ 2$ coins, and $n$ of them are made of silver. The coins are set at random in $n$ rows, with each row having $n$ coins. How do we calculate the probability that there is a row in which ...
1
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1answer
22 views

Random variable and distribution - number of tests a teacher has to make

$100$ students do a test. The probability of failing the test is $0.6$, those that failed, do a retest, the probability of failing the retest is $0.5$. Those that fail the retest do another retest. ...
0
votes
1answer
15 views

Reason behind convergence in probability definition

A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$ $$\lim_{n\to\infty}\Pr\big(|X_n-X| > \epsilon\big) = 0$$ But ...
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0answers
15 views

pdf for the sum of squared iid normal random variables

I am trying to find the distribution/pdf for the sum of squared $X_i$ iid observations from the normal distribution $X_1 ,..., X_n$ ~ $N(\alpha , \alpha)$, i.e. $X_1 ^2 + X_2 ^2 +...+ X_n ^2$. I was ...
0
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0answers
16 views

gaussian process convergence

if I have a series of gaussian processes : ($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that ...
2
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1answer
22 views

Combinatorics/Probability unordered lists

I don't really understand these unordered lists problems such as... Q: John goes to a store and buys 10 pieces of fruit from the selection of apples, bananas,peaches and pears at random. What is the ...
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0answers
17 views

On the probability of singular matrices containing whole numbers

Today in class - my teacher was teaching determinants . He gave us problems to solve of various kinds , including various row - column operations and determinants properties. But one thing that ...
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2answers
29 views

Probabilty exam question

I would like some help with what direction to take in this question.I find it difficult to decide what rule I need to use when I read a question. Cars pass at an average rate of 1 every 10 seconds. ...
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1answer
29 views

Drawing 6 balls of different colours

Hi I have an exam on Monday and am doing a few probability questions. I have checked the mark scheme for the answer to the following question however the method isn't stated. Could someone please ...
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0answers
16 views

Compare 2014 to 1998, 2014 has a 90% chance of being warmer than 1998?

According to NASA, 2014 has a 38% chance of being the warmest year, 1998 has only a 4% chance of being the warmest year. 2014 or 1998 have a 42% chance of being the warmest year. Since I eliminated ...
2
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1answer
22 views

Conditional Probability of Sinking Ship Question

Question: Two ships. Ship A's missiles have an 80% probability of hitting its target, ship B's missiles have a 50% probability of hitting the target. It only takes one hit from a missile to sink a ...
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0answers
13 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula [on hold]

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
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2answers
20 views

Prove that markov chain is recurrent

I have the following markov chain : $S=\{0,1,2,3\}$ $p_{i,0} = q$ (if we are in one of the states $0,1,2,3$ we can return to $0$ with probability $q$) $p_{i,i+1} = 1-q , i\in\{0,1,2\}$ (if we are ...