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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9 views

Find upper bound of probability value using Chebyshev's inequality

Given density function of random variabel X is f(x) = 1/(2√x), for -√3 < x < √3. Use Chebyshev's inequality to find upper bound of probability value P(IxI≥3/2).
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0answers
6 views

Find lower bound of probability value using Chebyshev's inequality

Given density function of random variabel X is f(x) = 3x^2, for 0 < x <1. Use Chebyshev's inequality to find lower bound of probability value : 1. P(5/8 < x < 7/8) 2. P(1/2 < x < ...
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1answer
7 views

Find the probability P(x is even) of given cumulative distributive function

Given cumulative distributive function (CDF) $F(x) = 1 - (1/2)^{(x+1)}$ for $x = 0, 1, 2, ...$ Find the probability value P(x is even).
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0answers
10 views

The probability of two dependent events occurring

If you wish to calculate the probability that both of 2 dependent events A and B will occur and you draw a tree diagram with A and B as the first two branches and then A and B again as two branches ...
0
votes
1answer
11 views

matrix sampling and its rank preservation

Assuming matrix $X\in R^{m\times n}$ is row orthogonal of rank $m$. Then, if I construct a new matrix $Y\in R^{m\times t}$, whose columns are directly sampled from $X$ with or without replacement ...
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0answers
5 views

$ | \sup_{x \in [0,1]} | x - \frac{1}{k} \sum_{i=1}^{k} \mathbb{1}_{(-\infty,x]}(X_i)| - \max_{i}| Y_{k-1,i} - \frac{i}{k}|| \leq \frac{2}{k}$

Let $k \in \mathbb{N}$ and $X_1, \ldots, X_k$ be uniformly distributed random variables on $[0,1]$. With $Y_{k-1,i}$ we denote the $i$-th smallest nuber in $\{X_1, \ldots, X_{k-1}\}$. How can I prove ...
0
votes
1answer
14 views

Derivation of negative binomial distribution

Let $X, Y$ be geometric distribution where $ \mathbf P(X=k) = \mathbf P(Y=k) = (1-p)p^{k-1}$ for $k = 1, 2, 3...$ Using the convolution formula: $$\mathbf P(Z=z)=\sum_{n=1}^{z} \mathbf P(X=z) ...
0
votes
0answers
13 views

expectation calculation problem small problem

a Continuous, positive random variable X, whose PDF is proportional to $(1+x)^{-4}$, where $0<x<\infty$, determine $E(X)$ i tried to solve it directly by integrating from 0 to infinity to get ...
0
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0answers
5 views

Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
1
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2answers
21 views

expectation calculation problem

I got the answers for this and i know its 1.05 but the way it explains is very difficult to understand so im seeking for some help here. A system made up of 7 components with independent, identically ...
0
votes
2answers
14 views

Applying the basic formula for binomial distribution

I'm pretty confused on how this works. In my class my teacher states that: Let $X$ be a random variable with $S_X = \{0,1\}$. $X$ follows a Bernoulli distribution if $P(X = x) = p^x(1-p)^{1-x}$ for ...
-1
votes
1answer
15 views

Probability Binomial Distribution Question [on hold]

The quality control unit in a medical device company inspects 20 pacemakers each hour. Let X represent the number of pacemakers in the sample of 20 that require rework. Pacemakers are assumed to be ...
-1
votes
0answers
15 views

The moment-generating function of a random variable X is given by the following formula [on hold]

$M(t) = c(e^t + e^{2t}){e^t + e^{−t}} + 0.6e^{4t}$ , $c > 0$. (a) Find the value of c where c > 0. (b) Find all possible values of $X$ and the corresponding probabilities. (c) Find $E(X)$. ...
0
votes
0answers
22 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...
-1
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0answers
23 views

Challenge Problems [on hold]

This question might be better fit for meta, but how might I find a list of challenge problems similar to the following. In addition this question may have already been asked. ...
1
vote
1answer
19 views

Minimum of random exponential variable and time

Let $U$ and $V$ be $\sim\mathrm{Exp}(\lambda)$. Let $s \in [0, t)$. Does this reasoning from left to hold: $P(U + (V \wedge t) \leq s) = P(U + V \leq s; V < t) $ or is in this case enough to write ...
0
votes
2answers
24 views

De Méré paradox. Show that throwing at least one “one” of 4 dice is more probable than throwing at least two “ones” of 24 tosses of a dice.

De Méré paradox. Show that throwing at least one "one" of 4 dice is more probable than throwing at least two "ones" of 24 tosses of a dice. Finding the probability of the first is easy : ...
0
votes
2answers
30 views

Calculating the expected value of a random variable that's a function of a random variable

I am working on the following problem: I'm having a hard time putting all of this information together: The cost of the maintenance is $Z = X + Y$, where $X$ is the cost of the first machine and ...
1
vote
1answer
13 views

What would the expected number of swaps in a merge sort be?

If I were given a list of random numbers say x1, x2, .........., xn and these numbers are sorted according to the merge sort algorithm. What would be the number of expected swaps/exchanges which would ...
2
votes
1answer
28 views

Expected Value and variance of a max randomized stocks

Hey guys I have been working on a probability and expected value/variance problem and the problem is: Each day the price of a stock in the market is a random number between 0 and 1 independently of ...
0
votes
1answer
30 views

How to find $E[Y|X=1]$?

A fair die is repeatedly rolled. Let $X$ and $Y$ denote, respectively, the number of rolls required to obtain a $1$ and a $2$. How do I find $E[Y|X=1]$? edit: for using this I got 1*6 = 6 total ...
0
votes
0answers
14 views

Simple explanation of the result

For $x \geq 0$ and $ 0 \leq y \leq t$ I get the result that $P(E_t > x, A_t > y) = e^{-\lambda x}*e^{-\lambda y} $ For $x \geq 0$ and $ 0 \leq y \leq t$ I get the result that $P(E_t > x, A_t ...
0
votes
0answers
26 views

Linear least mean square problem

I have a prior distribution $\Theta$ with mean $1$ and variance $2$, and noise term $W$ with mean $3$ and variance $5$.I also have two different instruments to measure $\Theta$, each having ...
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0answers
9 views

Exercise on stationary measures.

This is a question from Durrett, exercise 6.5.4. Recall that $$ \mu_x(y) = E_x\left( \sum_{n=0}^{T_x-1} 1(X_n = y)\right) = \sum_{n=0}^\infty P_x(X_n = y, T_x > n)$$ is a stationary measure and ...
1
vote
3answers
34 views

Hello expected output (probability question)

I am working on a probability problem I tried finding the total net productivity days based on the amount of machines the factory has, so if there was 1 machine, there will be 29 days * 1 machine = ...
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votes
2answers
23 views

Permutations; group of 5 boys, 10 girls. What's the probability the person the 4th position is a boy?

Problem description: A group of 5 boys and 10 girls is lined up in random order -- that is, each of the 15! permutations is assumed to be equally likely. What is the probability that the person in ...
0
votes
1answer
24 views

Some men and women are randomly assigned seats at a round table and no two persons of the same sex are seated next to each other. Probability of this?

Four women and four men are assigned seats at random at a round table. what is the probability that no two persons of the same sex will be sitting next to each other?
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0answers
12 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
0
votes
1answer
19 views

probability of tail event using kolmogorov's 0-1 law

If $X_1,X_2,... $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I ...
0
votes
1answer
44 views

Find $P(A\Delta B)$ from $P(A)$, $P(AB)$, and $P(A^cB)$

a) If $A,B \in \mathcal{F}$ occurrences that satisfy: $P(A)={1\over 6}$, $P(AB)= {1 \over 12}; P(BA^c)={1\over 18}$. Find: $P(A\Delta B)$ b) Prove that for $A,B,C,D \in\mathcal F$ the following ...
3
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1answer
35 views

Two numbers are chosen at random over the interval $ [0,1]$

Two real numbers, $x$ and $y$ are chosen at random over the interval $ [0,1]$. What is the probability that the closest integer to $\frac{x}{y}$ will be even? Floor functions don't place nicely with ...
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votes
2answers
57 views

I asked this question before , what somehow got deleted. I really need an answer so i ask again [on hold]

In a random fashion 3 letters of A, B and C are placed in a array. Find the probability that two of the same letters aren't neighbors in the array. Valid is: ABCBABCA, invalid would be AABCBCBCA. The ...
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votes
0answers
11 views

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

I think it has to be done by considering $A_n$ and $A_k^c$ separately.
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1answer
20 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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0answers
4 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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votes
3answers
14 views

Expectation of minimum discrete and continious random variable

I have two random variables, one is $X$ which has $\mathbb{P}(X=2)=\mathbb{P}(X=4)=\mathbb{P}(X=6)=1/3$, and one, call it $Y$, which is $unif(3,5)$ (continious) distributed. I want to compute ...
0
votes
1answer
16 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 ...
0
votes
1answer
25 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 ...
2
votes
2answers
87 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
27 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 ...
1
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1answer
25 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 ...
0
votes
1answer
30 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 ...
0
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0answers
25 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! ...
0
votes
0answers
19 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 ...
0
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0answers
10 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
17 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 ...
0
votes
1answer
38 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 ...
1
vote
1answer
47 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
24 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
votes
1answer
52 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 $\in \mathscr{G}$: ...