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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2answers
34 views

Probability $Pr(W<R)$ of two Normal Random Variables $W$ and $R$.

Let $R$~$Normal(μ_R,σ_R^2)$ and $W$~$Normal(μ_W,σ_W^2)$ Also $μ_W=4μ_R$ and $σ_W=4σ_R$ I have to calculate $Pr(W<R)$ or equivalently $Pr(\frac{W}{R}<1)$ I've got this far: Let ...
0
votes
2answers
43 views

Probability problem with combination of poisson and binomial distributions

Exercise The number of clients that enter to a bank is a Poisson process of parameter $\lambda>0$ persons per hour. Each client has probability $p$ of being a man and $1-p$ of being a woman. After ...
2
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1answer
30 views

Theoretical interpretation of simulating from a distribution

Suppose there is a random variable $X$ with marginal density $p_X$. However only the conditional densities $\{p_{X\mid\Theta}(\cdot\mid\theta):\theta \in \mathbf{T}\}$ are known directly, where ...
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0answers
25 views

Intuitive explanation of Shannon's source entropy in information / communications theory

I am trying to calculate the number of bits required to encode a message. FOr that, I am applying Shannon's entropy, H. I have done the implementation and playing around with themessage length and ...
0
votes
1answer
11 views

Limes superior and random variables

I want to show the following: Let $X_1,X_2\dots$ be i.i.d. random variables. Let $\text{E}[|X|^p]=\infty$ for $p>0$. Show that $$P(\limsup\limits_{n\to\infty }\{|X_n|\geq n^{1/p}\})=1$$ What I ...
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0answers
21 views

Question about notation for a statement about conditional probability distribution

Consider the random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, P)$. Suppose $Y$ is a discrete random variable with support $\mathcal{Y}\subset \mathbb{R}$. Suppose ...
1
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1answer
35 views

Determine probability of getting a desired outcome

Say there are n distinct outcomes and you have ranked each one from 1..n, where the higher the rank the more desirable the outcome. Now say that you are presented an ordered list of k such choices, ...
0
votes
1answer
30 views

showing independence of random variable and arithmetic mean

Let $X_1 , X_2 $ be iid random variables and $ \bar{X} := \frac{1}{n} \sum_{i=1}^n X_i $. I should answer the questions whether 1.) $ X_1 $ and $ X_1 - \bar{X} $ are independent 2.) $ X_1 - ...
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0answers
13 views

A rate model for sodium channels

I am studying by myself Human Physiology. I have encountered the following question: In the following given model of sodium channel with 3 states open closed blocked (which I assume means ...
2
votes
3answers
64 views

Probability - die - The number of throws until a $5$ and a $6$ have been obtained.

An unbiased die is thrown repeatedly until a 5 and a 6 have been obtained. the random variable M denotes the number of throws required. For example, for the sequence of results 6,3,2,3,6,6,5, the ...
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0answers
27 views

find $\text{limsup} \dfrac{X_n}{\ln{n}}$? how can i apply Borel-Cantelli here?

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{-|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
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0answers
19 views

Markov’s inequality Example

Let $X$ be a random variable which is always smaller than $1000$, and has expected value m. Show that for all $λ > 0$, $$\mathbf{P}(X < 100 - \lambda(1000 - m)) < \frac{1}{\lambda}.$$ ...
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1answer
19 views

Using 'At Least' in binomial probability [on hold]

Find the probability of at least one failure in 5 trials of a binomial experiment in which the probability of success is 30%
0
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1answer
24 views

If voters arrive according to a Poisson process, how can we find the conditional number of votes of a candidate?

Suppose that voters arrive to a voting booth according to a Poisson process with rate $\lambda = 100$ voters per hour. The voters will vote for two candidates, candidate $1$ and candidate $2$ with ...
0
votes
1answer
49 views

Infinite probability density?

I've read that for a "[..]random variable strongly "localized" around a single value", the probability density function (PDF) could be: $p(x)=\frac {1}{2\epsilon}$, with $\epsilon \to 0$, and ...
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0answers
24 views

Probability of rainfall [on hold]

There is rainfall in a certain place for 10 days in every 30 days. Find the probability that, i) there is rainfall on at least 3 days of a given week and ii) the first four days of a given week will ...
0
votes
1answer
29 views

Finding distance from valid number

I have a game related problem that is pretty complex. Here is the simplified version of the problem. I have a list of "good" numbers. ...
0
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1answer
11 views

A bound for error function

I am looking for a bound (or dominated function) of $erf(x)$ where $erf$ is defined here https://en.wikipedia.org/wiki/Error_function Thank you very much.
0
votes
1answer
42 views

Multiple Choice Test with 240 questions [on hold]

I have a multiple choice test with 240 questions, the pass rate is 60% and there is 3 options for each question. What is the probability/percentage of passing the test by simply guessing each ...
0
votes
1answer
28 views

Basic probability oak and pine question

The probability that an oak log is split on any chop is $1/3$ and the probability that a pine log is split on any chop is $2/3$. In a large batch of logs, three quarters of the logs are oak and the ...
0
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0answers
12 views

Formula for the minimum number of trials for statistic to match probability

A fair dice has an x% chance of coming up 1. It is rolled y times and results written down. What is the lowest value of y such that the odds of exactly x% of the written results being 1, are better ...
0
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0answers
20 views

Corelate Rating and Confidence

Imagine a one evaluation list composed by two variables: Rating and confidence. Rating and confidence are in the scale of 0-100. For a specific combination of Rating and confidence i need to achieve ...
0
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0answers
29 views

Is $ \mathbb{E}f(X)g(X) \ge \mathbb{E}f(X)\mathbb{E}g(X) $ always true?

Assume that $f$ and $g$ are non-negative functions. The inequality is true when f=g=identity (using the non-negativity of variance). Is it true in general?
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2answers
42 views

Probablity of Coin flipping.

I Have 2 Coins- The first is a fair coin - containts Head/Tail. The second is not a fair coin - containts Head/Head. I chose some coin randomaly and flipped it, and got Head. What is the probablity I ...
0
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2answers
35 views

Find dependent event when two dice are thrown simultaneously.

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of ...
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0answers
8 views

Scotts axiom, representation theorems for Qualitative -Numerical Probability function relations

Scotts theorem/axiom and other representation theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, ...
0
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0answers
42 views

Roll dice and get score except when you get 6 where you lose everything [closed]

At what score to stop when you roll a dice under these conditions: -if you get 6 you lose all your score -You score is the add up of all rolls
0
votes
1answer
15 views

Covariance vs Pearson Correlation Coefficient

Assume that is given a Gaussian Distribution with positive definite covariance matrix $\Sigma$. For some off diagonal elements we have $|\Sigma_{i,j}|>|\Sigma_{i,k}|>|\Sigma_{i,l}|>...$ . Is ...
0
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0answers
27 views

Waiting Time Distribution

Let X be a random variable which denotes the amount of time spent in a state(say state 'I') before changing state. As X is a random variable it must have a Probability space/sample space and a sigma ...
2
votes
1answer
39 views

If you choose 4 of these lamps at random, what is the probability that none need to be replaced during the first $150$ hours of use?

The lifetime, in hours, of each lamp produced by a certain company, is a random variable with density function given by $$f(t)=\begin{cases}100/t^2,& t>100\\ ...
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0answers
40 views

Probability problem of fishes in a lake

Exercise In order to estimate the number $N$ of fishes in a lake, a fisherman executes the following procedure: in the first step, he captures $n$ fishes and after marking them, he returns them to ...
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votes
1answer
35 views

Use a $Z$ table to find $P(-1 < Z < 1)$. [closed]

Can someone help with these problems, please? Use Appendix Table III to determine the following probabilities for the standard normal random variable $Z$: (a) $P(-1 < Z < 1)$ (b) $P(-2 < Z ...
0
votes
1answer
28 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
2
votes
1answer
25 views

probability problem with Poisson distribution

Problem A retailer knows that the demand of boxes is a random variable with Poisson distribution of parameter $\lambda=2$ boxes per week. The retailer completes his stock on monday so as to have four ...
1
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1answer
26 views

Type 2 Error Question - How to calculate for a two tailed?

The modulus of rupture (MOR) for a particular grade of pencil lead is known to have a standard deviation of 250 psi. Process standards call for a target value of 6500 psi for the true mean MOR. For ...
1
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2answers
59 views

Bayesian urn questions

There are two urns, each with four ping-pong balls. In one urn, three of the balls are red, and one is white; in the other, three are white, and one is red. Without knowing which urn you are choosing, ...
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3answers
86 views

Winning All Levels in a Game

There are $L$ levels in a game. In each turn of the game, you go through each level one by one and try to complete it. The goal is to complete all levels of the game. The probability of completing any ...
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0answers
21 views

Determine p value of the statistics [closed]

Given X is a geometric random variable with pmf and cdf $$p(x) = 0.153(1-0.153)^{x-1} $$ and $$F(x) = 1-(1-0.153)^x$$ A statistics S is defined by $S = \min\{X_1, X_2,X_3,\ldots,X_n\}$ If the ...
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0answers
40 views

Facebook Data Science Question (Expected Payout and Probability)

I saw this question on Glassdoor and couldn't seem to find a answer to validate mine anywhere: You're at a casino with two dice, if you roll a 5 you win, and get paid $10. What is your expected ...
0
votes
1answer
35 views

How to prove MLE of theta is unbiased?

Let $X_1, X_2, . . . , X_n$ be a random sample from a uniform distribution on $[0, \theta]$. Suppose results $x_1, x_2, . . . , x_n$ are observed. Since $f(x) = 1/\theta$ for $0 \leq x \leq \theta$, ...
1
vote
2answers
56 views

Expected number of coin tosses

A fair coin is tossed until either 4 heads or 9 tails obtained ( total). What is the expected number of tosses? Edit: I calculated the probability. It is 10% for 4 heads and 12% for 9 tails. But how ...
0
votes
1answer
33 views

Uniform Distribution Question - Help Needed

Let $X_1, X_2, . . . , X_n$ be a random sample from a uniform distribution on $[0, \theta]$. Suppose results $x_1, x_2, . . . , x_n$ are observed. Since $f(x) = 1/\theta$ for $0 \leq x \leq \theta$, ...
1
vote
2answers
31 views

Probability Question - Moment Generating Function

$$ f(x) = \begin{cases} xe^{-x}, & \text{x ≥ 0} \\ 0, & \text{elsewhere} \end{cases}$$ Q: Find the Moment Generating Function of X. Hi, I was trying to solve this question by putting the ...
2
votes
3answers
64 views

Picking two random points on a disk

I try to solve the following: Pick two arbitrary points $M$ and $N$ independently on a disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2 \leq 1\}$ that is unformily inside. Let $P$ be the distance between those ...
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0answers
29 views

how find an event with probability $6.6p^2q$ [closed]

How to find an event with the following probabilities $6p^2q$,$6.6p^2q$,$6.75p^2q$,$3.9pq$,$4pq$ using independent bernoulli trials?
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0answers
20 views

Posterior of Normal with prior Cauchy

Let $X\sim N(\theta,1)$ and $\pi(\theta)\sim \mathrm{Cauchy}(0,1)$ find a 90% credible set for $\theta$ To find the credible set I need to find the distribution of $f(\theta\mid x)$, but ...
2
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0answers
29 views

Can someone help me balance this game (probability question) [closed]

A team of 9 vs a team of 1. Each round each of "the 9" roll a die to "attack" and "the 1" rolls 9 dice to "defend", the nine dice are preassigned to attackers before the roll, "the 1" cannot choose ...
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votes
1answer
12 views

How do i find the Maximum Likelihood Function Estimator [closed]

Im trying to find the estimator for $\mu$ but the lecture notes i have don't explain very well. The question is find the maximum likelihood function estimator of: $$L(\mu : x) = c. ...
-1
votes
2answers
39 views

Finding the PDF of Y=X-2 [closed]

I am given the following PDF of random variable $X$: $$f(x)= \begin{cases} e^x & \text{for }x<0, \\ 0 & \text{otherwise}. \end{cases}$$ a) Compute $E(e^x)$: Here is my work: ...
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0answers
18 views

Change of variable formula for density function

We all are aware of the change of variable formula whereby if $$[A, B] = g(X, Y) $$ and g is invertible, then the joint density function of A, B is given by $$f_{ab} (A, B) =1/|J| f_{XY} (g^{-1}(a, ...