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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4 views

For sequence of events $\{A_n\}$ in probability space, show that $\lim_n P( \lim\inf_k A_n \cap A_k^c)=0$

I think it has to be done by considering $A_n$ and $A_k^c$ separately.
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1answer
12 views

Example of strict inequality in special case of fatou's lemma.

Give an example of sequence of events $\{A_n\}$ such that the following inequalities are strict $P(\lim\inf A_n) \le \lim\inf P(A_n) \le \lim\sup P(A_n) \le P(\lim\sup A_n)$. Thanks
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3 views

A mix between the Horvitz-Thompson and ordinary estimator

I have asked this question on mathoverflow, but got no answer. Here I have corrected some mistakes and wish to hear any ideas that may bring at least numerical result: The data I have two samples: ...
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1answer
13 views

Selection of Distribution model

An expressed parcel delivery company offers a First Class service for which it is promised that 80% of all parcels are delivered within 24 hours of dispatch. It is suspected that the true successful ...
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1answer
20 views

Finding Variance

I am a little confused on how to go about finding different parts of the Variance of a random variable. Here is the question. A total of $n$ balls, numbered $1,.. n$, are put into $n$ urns, also ...
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0answers
40 views

$X_1, \dots, X_n$ are independent random variables. Suppose $M = \min(X_1, X_2, \dots, X_n)$

Given that $X_1,\dots, X_n$ are independent random variables. Suppose $M = \min(X_1, X_2,\dots, X_n)$ and $X_i$ are exponential random variables with parameter $λ_i$, compute $E[M X_j | M = X_i]$ ...
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0answers
24 views

understanding darts probability

Note: this problem for who understands the game of darts Hello iam trying to compute the probability of a dart to hit a ring if you know that the opportunity to miss the ring is 10% what will the ...
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0answers
17 views

Percentage of failed devices.

According to one of the Western Electric rules for quality control, a produced item is considered conforming if its measurement falls within three standard deviations from the target value. Suppose ...
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0answers
17 views

formal proof that p-values are uniformly distributed

I'm trying to prove that $p$-values under the null hypothesis are uniformly distributed in $[0, 1]$ for an absolutely continuous test statistic $X$. Proof: By continuity of $F_X$, it is sufficient to ...
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0answers
23 views

Conditional distribution of two binomials which both depend on a third

I have a question that I'm having some trouble with, but which I believe might have a fairly straightforward answer. I'd really appreciate it if someone could help point me in the right direction! ...
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0answers
18 views

How to get more profit in stochastic process?

Suppose there is a system, for each step, I cost something but I didn't know how much I cost, and the system return to me something, which follow Guassian distribution and the expectation is what I ...
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0answers
8 views

Book recommendation needed: asymptotic behavior of non-stationary Markov chain

Is there any stochastic process textbook which covers some standard results for non-stationary Markov chain? For my purpose, countable state space is enough. Thanks!
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1answer
13 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
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1answer
21 views

Variance and Expected value of internet connection

I am working on a probability/statistics problem! The problem is as follows: Your internet connection is very poor. It constantly alternates between being functional for x minutes and being down for ...
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1answer
44 views

How to find $E[X^2\mid X+Y]$?

Suppose $X$ and $Y$ are independent Poisson random variables with rates $\lambda_1, \lambda_2$ respectively, then how would we go about calculating: $ E[X^2\mid X+Y] \text{ ?} $$
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0answers
21 views

Probability - Gambling / Decision Trees

So this question is related to decision trees/probabilities/bayes theorem. Sorry that it's quite long, but exams/tests for this course have been basically 3-4 questions of this length. Danny goes to ...
2
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1answer
50 views

Prove Y = X given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$

Prove Y = X, given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$ Attempt: Suppose $Y = E[X|\mathscr{G}] $. Then $E[X|\mathscr{G}] $ is $\mathscr{G}$-measureable. For every A $\in \mathscr{G}$: ...
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3answers
418 views

High-School level probability and logic problem

So the other day I took a math test, (not for class, its just an optional test, so this isn't and kind of cheating) which included all kinds of logical and problem solving exercises, among others this ...
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1answer
26 views

Uniqueness of Spatial Median

https://projecteuclid.org/download/pdf_1/euclid.aos/1176350511 Can You help me understand why there is less-than sign in the proof? ...
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1answer
41 views

How long would it take to a lottery number repeat?

In Professor Stewart’s Cabinet of Mathematical Curiosities the following is asked: You have $1000$ songs on your MP3 player. If it plays songs ‘at random’, how long would you expect to wait ...
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1answer
43 views

Find probability of $X\leq Y$

I have two random variables, $f_ X(x) = \left\{ \begin{array}{ll} \frac{2}{x^3}, & \mbox{if $x \geq 1$}, \\ 0, & \mbox{otherwise}. \end{array} \right.$.$f_ Y(y)=\frac{1}{2},$ uniform ...
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2answers
55 views

A dice is tossed 500 times, what is the probabioity of ‘1’ coming up 100 times? [on hold]

A dice is tossed 500 times, what is the probabioity of ‘1’ coming up 100 times?
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1answer
27 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
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2answers
19 views

Prove that if $k\mid n$ then $p(A_k)={1\over k}$

Let $n$ be a natural number, $n=p_1^{a_1}\cdot...\cdotp_m^{a_m}$. Let us randomly choose a number between 1 and $n$ with a uniform, equal chance. Let us denote the event "The number chosen is ...
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1answer
38 views

Dice Probabilities - How often does 3, 4, 5 show up in 5 dice on one roll?

I want to find out the probability for rolling 5 dice and the result containing at least 3, 4, and 5 without caring what the other 2 numbers are. I'm not very familiar with this kind of probability so ...
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0answers
25 views

Probability_distribution [on hold]

Three points are chosen at random on the circumference of a circle. Find the probability that they all lie on the same semicircle, using random numbers generated from a uniform distribution.
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1answer
24 views

Bayes' theorem - Updating Probabilities

Thanks for reading this. I'll caution that it's quite the wall of text, but this is for my stochastic processes course. Due to recent news of horse meat in frozen food products, ABC Meat Distributors ...
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0answers
25 views

4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose.

Question: 4 cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You win if you turn over ...
2
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0answers
45 views

incorrect rejection of a true null hypothesis?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
0
votes
3answers
27 views

Probability of success in $n$ trials

I'm stuck on my statistics homework and would appreciate your help. Question: Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is $0.12$. ...
0
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1answer
24 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; ...
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1answer
22 views

Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
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1answer
44 views

What is the probability that you get $i$ on the $i^{th}$ trial?

What is the probability that you get $i$ on the $i^{th}$ trial? Match = Get result $i$ on $i^{th}$ trial. What is the probability of $M = 0,1,2,...,6$ matches when: Note: I'm not asking you to do ...
0
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1answer
21 views

Expectation of two dependent normal distribution

$U,V,W$ are independent normal distributions with $0$ mean and unit variance. I am given $X=U+W, Y=U+V$.How can I find $E[XY]$? I know $E[X]=E[Y]=0$, But clarely, $X$ and $Y$ are not independent ...
2
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0answers
19 views

An inequality for symmetric random walk

I need to show that if $(X_j)$ are symmetric i.i.d. random variables with partial sums $S_n:= \sum_{j=1}^n X_j$, then for all $x \geq 0$ $$P(|S_n| > x) \geq \frac{1}{2} P(\max_{1 \leq j \leq n} ...
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2answers
44 views

How to score a second try on a multiple choice test

Suppose a multiple choice question has $n$ answers. You allow a student $m$ tries. The student gets the correct answer on the $k$-th try. What is a "fair" score?
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2answers
31 views

Fair dice probability problem

You flip a coin 7 times and observe 2 heads and 5 tails. Calculate the probability that the event that you observed might actually come from a fair coin.
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1answer
17 views

Application Birkhoff ergodic theorem

Let $(X,\mathcal{B},m,T)$ be a probability preserving transformation. Let \begin{align*} I:&=\{f\in L^1: f=f\circ T\}\\ B:&=\{g-g\circ T: g\in L^1\} \end{align*} I have to show that $$ ...
0
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3answers
21 views

An ice-cream parlour offers 15 different flavours of ice-cream. An ice-cream sundae contains 5 scoops of ice-cream.

An ice-cream parlour offers 15 different flavours of ice-cream. An ice-cream sundae contains 5 scoops of ice-cream. Suppose someone selects the five scoops of a sundae at random (repetitions allowed). ...
1
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1answer
19 views

How many possible different telephone numbers consisting of two zeros, three fours, one five and one seven are there?

In a city every telephone number has 7 digits where the first digit is never a 0. How many possible different telephone numbers consisting of two zeros, three fours, one five and one seven are there? ...
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1answer
10 views

Bounds-negative binomial distribution

Suppose $Y=\sum_{i=1}^{n} X_{i}$ where each $X_{i}$ is an independently and identically distributed geometric random variable with success parameter $p$, so that $Y$ has a negative binomial ...
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0answers
35 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...
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0answers
28 views

What does u mean in density function [on hold]

I have such hometask - We have a density function: $$ f_X(x) = \frac{1}{2} u(x)\,\exp\left(-\frac{3x}{2}\right) $$ u is a step function So I need to find expected value of $X^3$. How to solve this?
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1answer
28 views

Representing the probability as a recurrence equation

Introduction Suppose that you initially have an $n$-sided die with equal probability and you throw it then you will get a certain number $1< k \leq n$ then you throw a $k$-sided die. Continue ...
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1answer
31 views

Is this Event Mutally Exclusive?

I am trying to calculate the following, however I'm unsure on whether this event would be Mutally Exclusive or Independent. Can someone help with finding the probability of the Intersection? P(A) ...
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1answer
17 views

Distribution with the following moment generating function?

Can anybody suggest a distribution whose Laplace transform is the following? $\mathrm{E}[e^{tX}] = \exp(\lambda(e^{2t}-1))$. Note: The MGF of Poisson distribution is $\exp(\lambda(e^{t}-1))$.
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1answer
19 views

Let $G$ be a random variable with $G=G(X,Y)$ where $G(x,y)=3x+y^2$. find $E(G)$

Let $G$ be a random variable with $G=G(X,Y)$ where $G(x,y)=3x+y^2$. Find $E(G)$ i know that $E(X)=E(Y)=1$ and $Var(X)=Var(Y)=1/2$ is this enough to answer the question? Im sure this is'nt too hard ...
0
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1answer
21 views

Which way to calculate expectation is correct?

Might be a stupid question, but I really got stuck. Suppose, there is some ex-ante unknown amount of money $c$ in your pocket. It is drawn from some distribution $F$ over $[0,N]$. If the amount ...
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0answers
13 views

probability distribution log normal [on hold]

is the distribution of log normal of x uni modal? I have tried to form a pdf of a log normal function of x, I need to know if the distribution is unimodal
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1answer
25 views

Uniformly distributed independent random Variables [on hold]

Let X and Y be independent random variables each uniformly distributed on (0,1). Find $P(Y\geq X | Y\geq \frac{1}{2})$. The answer is $\frac{3}{4}$ But I don't know how they got it :( Please help as I ...