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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1answer
12 views

Approximating distributions by finite number of moments

Let $P$ be a distribution that is moment-determinate, i.e. it has finite moments of all orders and the infinite moments determine $P$ uniquely. I was asking myself whether there are results on the ...
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0answers
18 views

A Markov Chain Problem.(Change the color of ball)

There are $n$ different color balls in a box. Take two balls in turns, and change color of the second ball to the first. (This is one operation). Let $k$ be the (random) number of operations needed to ...
4
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3answers
26 views

Strategies to guess choices for multiple choice questions

Multiple choice questions (MCQs) are common in examinations over here in Singapore. A set of, say, $40$ questions are given to students, and each is accompanied with a list of $4$ choices of ...
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4answers
26 views

Probability: Linear Seating Arrangement

Okay, I'm new at probability and statistics, so please try to answer this as thoroughly as possible and explain why you did everything, from using a specific number to why using factorials and ...
0
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3answers
67 views

Is it true in general that $E(1/X) = 1/E(X)$?

This concerns a discrete random variable $X$. I assume the relation doesn't hold in general, but I would like to prove this. I have tried to use the property that $$ E(g(X)) = \sum_x g(x)f(x) $$ ...
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2answers
259 views

What is the probability of my sum reaching exactly 10?

I throw a 6-sided dice (with values: 0,1,2,3,4,5) multiple times and add each value to a sum, which is 0 in the beginning. What is the probability of my sum reaching exactly 10, 11, 12, 13, 14? After ...
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1answer
24 views

Meaning of 'expected value' in the following problem

Ok, I have found an interesting probabilites problem on TopCoder. I have truncated the statement: "What is the expected number of dice throws needed to attain a value of at least n (candies, in this ...
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2answers
15 views

Calculating Probabilities for a cumulative distribution function within a given inequality

Given that K = 1/36, I require some help understanding (b) • Pr(1/2 ≤ X ≤ 1) Is re-written as such: Pr(X ≤ 1) - Pr(X < 1/2) I do not understand why! Is it because Pr(X ≤ 1) is solved as ...
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3answers
57 views

How to use stars and bars(combinatorics)

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$ Where $x_i\in\mathbb{N}$ Is this the correct time to apply the method?
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1answer
12 views

Calculating Probabilities using a cumulative distribution function

For (b) Pr(X greater than or equal to 2) = ? The textbook says as such but I am confused: Pr(X greater than or equal to 2) = 1 - pr(X less than 2) I do not understand why they re-write the ...
0
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1answer
18 views

A Borel-Cantelli lemma exercise.

Suppose ${A_n}$ is a sequence of events. If $P(A_n)\to 1$ as $n\to\infty$,prove there exists a subsequence ${n_k}$ tending to infinity such that $$P(\cap_kA_{n_k})>0$$ The textbook gives a hint ...
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0answers
5 views

How to derive the gradient formula for the Maximum Likelihood in RBM?

I am learning RBM (restricted Boltzmann machine) for deep learning. The log-likelihood of RBM is given as : and its gradient w.r.t. the parameter is: I don't understand how is the gradient derived ...
0
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1answer
15 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
1
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1answer
23 views

Can we find two measures $\nu$, $\mu$ which $\nu\ll\mu$ and $\mu$ is $\sigma$-finite while $\nu$ is not $\sigma$-finite?

Can we find two measures $\nu$, $\mu$ which $\nu\ll\mu$ and $\mu$ is $\sigma$-finite while $\nu$ is not $\sigma$-finite? I want to justify the Radon-Nikodym theorem but couldn't find an example.
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0answers
36 views

Probability: the average times to make all the balls the same color

Suppose there are n balls with different colors with each other in a bag. In one loop, One take two balls in sequence out of the bag and replace them with two balls with the same color of the first ...
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3answers
166 views

Cards in box - probability a given type is picked last

I came out with a probability question which I find difficult to solve. I hope some kind souls can provide me with some ideas. There is a box with four different types of cards, namely A, B, C, D. ...
0
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1answer
28 views

On showing that a given set is an Algebra

I'm having trouble on showing this: $\bullet$)We say a set $A$ $\subset$ $\mathbb{Z}$ is a periodic set if exists an integer $i$ and a set $I$ $\subset$ $\{1,\cdots,i\}$ such that: $$A = ...
2
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1answer
25 views

Chance that one of the beans in this jar is not black?

Nothing fancy here, I am just looking for an intuition check. Jar has $1000$ beans in it. You don't know anything about the beans in the jar or where they came from. You are allowed to shake the jar ...
0
votes
1answer
41 views

What is the mean and variance of $Y$, where $Y$ is sum of iid's

Here's my work for part a. I could use clarification on part b and d. Is part d the same as part a ($E[A_n] = E[Y]$) ? a) $$E[Y_n] = E[\frac{X_n}{2^n}]$$ ($X$'s are iid so...) $$= \frac{E[X]}{2^n} ...
1
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1answer
59 views

A bag contains 4 balls. Two balls are drawn at random and are found to be white. What is the probability that all balls are white. [duplicate]

this problem comes under the topic Baye's theorem. I have no clue how to solve this. I found this answer from one book,but not sure whether its correct
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1answer
23 views

Expected value expressed by CDF.

I have found following formula for expected value: $$\operatorname{E}[X] = \int_0^\infty \int_0^x \! \mathrm{d}t \, \mathrm{d}F(x) = \int_0^\infty \int_t^\infty \! \mathrm{d}F(x)\mathrm{d}t = ...
0
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1answer
20 views

Simple CDF question

What are the steps needed to turn the LHS of the following equation into the second and third steps in the following equation: Let Z be a RV with standard normal distribution. Then $Pr(|Z|\le x)\ge ...
1
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1answer
14 views

Avg # of Rectangle Intersections in 2D Field

So imagine I have a large 2D field. Thousands of small rectangles overlay the field. The field is much larger than the rectangles. The rectangles are placed randomly in the field such that they may or ...
1
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1answer
21 views

Markov property for discrete Markov chains. A question about “adjacent random variables”

Consider a discrete Markov chain (with values in $\mathbb R$) $\{X_n:\, n\in\mathbb N\}$: namely the state space $S$ is a countable subset of $\mathbb R$ and the random variables are $X_0, X_1, ...
2
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2answers
55 views

Probability of events in an infinite, independent coin-toss space

I am studying Steven E. Shreve's Stochastic Calculus book. Example 1.1.4 (p.4-6) constructs a probability measure on the space of infinely many coin tosses $\Omega_\infty$. In the example the ...
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2answers
37 views

What is the weather probability?

I checked similar threads and understood one thing from them, but I need some help with this problem. "A newspaper presents the weather in this form. Linn, Maria and Victoria discuss the weather ...
2
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1answer
40 views

A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
0
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1answer
19 views

A problem about indefinite integral in measure theory

tirple$(\Omega,\mathcal{A},P)$ Suppose $\xi$ is a random variable.Indefinite integral$$\varphi(B)=\int_B\xi\mathbb{d}P \quad\forall B\in\mathcal{A}$$ I saw in a textbook: If $E(\xi)$ exists(not ...
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0answers
21 views

SAMPLE SPACE CALCULATION [on hold]

Items coming off a production line are marked defective or nondefective. Items are observed and their conditions listed. This is continued until two consecutive defectives are produced or four items ...
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2answers
10 views

Expected Profit for Binomial Variable

Part (a) I am familiar with: (a) P(batch is rejected) = P(X greater than or equal to 3) and n = 15 and p(defective) = 0.1 This gives me the correct answer of 0.1841 I am stuck at part 2! I have ...
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1answer
28 views

Positive-definite + continuous at 0 $\Rightarrow$ continuous?

Let $F$ be a functional from $L_2(\mathbb{R})$ to $\mathbb{C}$ that is positive-definite*. We also know that $F$ is continuous at $0$. Can we deduce that $F$ is continuous over $L_2(\mathbb{R})$? ...
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1answer
24 views

probability question3 [on hold]

1)given S={e1,e2,e3} does there exist a number a so we get a probabiliy model: p{e1}=2a, p{e2}=a-1/2 , p{e3}=a? 2) S={e1,e2,e3,e4}, calculate p{e3} i) p{e1}=0.3, p{e2}=0.1, p{43}=0.2 ii) p{e1}=0.1, ...
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3answers
41 views

Calculating expected value for a Binomial random variable

How do you calculate $E(X^2)$ given the the number of trials and the probability of success? $E(X) = np$, then $E(X^2) = $? Do we have to draw up a table for $n=0,1,2,\ldots,n$ and then use the ...
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0answers
34 views

A problem with total probability theorem. [on hold]

A child, who has both parents suffering from allergies, has a risk of 60% of it in turn, but this probability is reduced to 40% if a single parent has allergies. surveys carried out at national level ...
0
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1answer
46 views

Find the probability of hotel being occupied, given the dependence on weather conditions

Problem The owners of a hotel in a popular ski resort are deciding the amount of supplies for the winter. Based on the collected data relating to previous years have determined that the probability ...
6
votes
2answers
426 views

Find the probability of winning at this lottery.

So, the problem I found goes like this: You have $n$ different numbers, numbered from $ 1 $ to $n$. You can randomly choose $m$ (different) of them. The computer also randomly selects $m$ ...
0
votes
1answer
12 views

Discrete Uniform Distribution SOA Practice Problem

X has a discrete uniform distribution on the integers 0,1,2,...n and Y has a discrete uniform distribution on the integers 1,2,3,...n. Find Var[X] -Var[Y] the answer in the book is $ ...
2
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3answers
262 views

Probability of dying from smallpox?

A family of four is infected with Variola major. There is a fatality rate of 30%. Calculate the probability that... Here are my attempts, The probability that nobody dies, ...
2
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2answers
35 views

Coupon Collector Prob Variation

I a programmer not a mathematician so please excuse my ignorance and please dumb it down for me. My research indicates that this is a variation of the Coupon Collector problem but I really don't know ...
0
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0answers
19 views

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ [on hold]

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ Can you help me with tips and bibliography... I don't understand very good the topic, and I can't ...
0
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1answer
33 views

A Probability Game of Bridge Problem and Its Puzzling Correct Answer

In my effort to work on my probability and combinatorics skills before the semester starts, I have come across this problem: What is the probability that a bridge hand will contain 13 cards of the ...
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0answers
24 views

Parental Markov Condition Example

I'm currently reading a text on Bayesian networks and the text is giving some very crude interpretations of what appear to be some of the most important foundations of the subject. It states the ...
0
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1answer
27 views

Calculate the overall chance of something happening in a trial

I've searched everywhere for an answer but couldn't find a formula to use. I have an independent event that has a 1% chance of happening every second. In my trial, there are 30 seconds. How can I ...
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0answers
33 views

Why does $p(x) = \int d\theta \ p(\theta, x) = \delta(x-X)$

I am reading a probability book and at some point, the following equation comes up: $$p(x) = \int d\theta \ p(\theta, x) = \delta(x-X) $$ where $\delta$ is the Dirac delta. Why is this true? I ...
1
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1answer
21 views

Number of events in Poisson process, nondisjoint time intervals

Let X,Y be the number of 'successes' in a poisson process with parameter $\lambda$ in the time intervals $I_1,I_2$. Compute the expectation $E(XY)$. If $I_1, I_2$ are disjoint then it is simply ...
0
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2answers
27 views

How many ways are there to win on exactly 3 out of 10 lottery tickets?

This question is for my own understanding of probability. Suppose that for a particular lottery ticket game, the odds of winning a prize are 1 in 4.66. I want to know my chances of winning on ...
2
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0answers
26 views

Probability in Medical Testing [on hold]

This is a real life question: A medical diagnostic test, Test 1, will test if a certain infection is present. The test will be positive or negative. Whichever the result, the probability that the ...
3
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0answers
53 views

Understanding probabilities in a puzzle solution

I'm having a problem understanding a solution based on probabilities in the following puzzle: Puzzle: There is a "triangular" duel between the three shooters. Everyone shoots one by one, can shoot ...
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0answers
45 views

Law of large numbers weak vs strong

Does someone have an example where the strong law of large numbers do not hold, but the weak law do hold ? If you think there is no such example, please explain why there are 2 laws of large numbers ...
2
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2answers
34 views

How to maximize pay with repeated toss of coin

repeated toss a coin and you can stop anytime and payoff is just #times you got head divided by total number of throws, how do you maximize your pay. Does anyone have a clever strategy for this? ...