Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. For questions about the theoretical footing of probability (especially using [tag:measure-theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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2answers
56 views

Simple yet somehow hard Statistical Hypothesis Testing

I am trying to understand Statistical Hypothesis Tests. I have a dice that gives a six with probability 1/6 and I want to see if it really is the case. $$H_0: \ p=\frac{1}{6} \ \ \ \ \ \ \ \ \ H_1: \ ...
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16 views

Estimate the smallest trace value among permutations of a random matrix

For a random matrix $X \in \mathbb{R}^{n\times n}$ where all elements follow normal distribution $X_{i,j}\sim \mathcal{N}(0,\sigma^2)$. Is there any way to approximate the following quantity? \begin{...
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30 views

$ Cov(X,Y)=-5, \, Cov(X,Z)=2, \, Cov(Y,Z)=2 $. What can we say about $\sigma^2 (X+Y+Z)$?

I found this question in some exam, but i don't have the answer. I tried to solve it myself, but i'm afraid i got stuck. We these random $3$ variables $X,Y,Z$, and we know that: $$ Cov(X,Y)=-5, \quad ...
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0answers
88 views

The probability of mistake in a program

Consider a program to detect if input number is prime or not: it returns true if the input number is prime it returns true with probability of $1/4$ if the number is not prime How many times you ...
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1answer
88 views

Tails of the normal distribution

I'm currently reading Roman Vershynin's High-Dimensional Probability. For Proposition $2.1.2$, I wonder how the lower bound is obtained. I understand that the lower bound is correct, but I don't know ...
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0answers
15 views

Concentration bound of binomial distribution $B(n,p)$ with bounded $p$

Let $X$ denote a random variable that follows the binomial distribution $B(n,p)$ where $p\in [a,b]$. How to upper bound the probability $P\big[|X-E[X]|>\frac{1}{2}E[X]\big]$?
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19 views

Notation clarification from the Wikipedia entry for the Law of Total Expectation

In the "Proof in the finite and countable cases" section of the Wikipedia article for the law of total expectation here it is written that: Let the random variables $X$ and $Y$, defined on ...
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18 views

Probability mass function of Pearson's chi-squared statistic

Recently , I encounter a problem about Probability mass function of Pearson's chi-squared statistic , I wonder the rate of n at the maximum point
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34 views

I want to know the expectation is satisfied.

I want to prove the below proposition. When $$\mathbb{E}_{x - p^{'}(x)}[\log({q(x) \over p^{'}(x)}+1)] \ge \mathbb{E}_{x - p(x)}[\log({q(x) \over p(x)}+1)]$$ is the below inequality satisfied? $$\...
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1answer
37 views

I draw 7 cards from a standard card deck, what is the probability 5 of them are the same suit?

Apologies for the formatting, I am not familiar with this platform. I believe I have solved it but I looking for assurance. Let X be a random variable indicating the number of cards of the same suit (...
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2answers
46 views

Expectation of maximum element of a vector whose expectation is zero

Informal statement: In short what I am asking is that we have a 2d vector whose elements are drawn i.i.d from $\mathcal{N}(0, 1)$. Then, we pick the maximum of the two. Once the index of the maximum ...
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1answer
63 views

Situations in which new information is irrelevant in conditional probability

I have a box containing 3 bags. Each bags contains three marbles. Bag A contains two yellow marbles and one red; Bag B contains two reds and one blue; Bag C contains two blues and one yellow. I ...
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1answer
21 views

Upper Bound on the Probability of the Difference of Binomial Distributions

We start by defining a binomial difference distribution Let $X\sim \text{Bin}(n, p)$, $Y\sim \text{Bin}(n, q)$, $Z=X-Y$ I've found out that this distribution is somewhat difficult to write down ...
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19 views

Analytically showing one correlation coefficient is stronger than another

Consider a set of finite $n$ pairwise-points of two variables, $X$ and $Y$ such that and their correlation coefficient. For simplicity, use the Pearson correlation coefficient. Consider another set of ...
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0answers
28 views

Expected vs observed average numbers

A manufacturer claims that a bag of jelly beans contains an average number of 312 sweets. A student buys 6 bags, counts the number of beans in each bag, and finds 309, 314, 310, 307, 311 and 312. Do ...
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2answers
28 views

Method of moments estimator for lognormal distribution

Let $X_1,\cdots X_n$ be identically and independently distributed lognormally. I want to find the method of moments estimators for $\mu,\sigma^2$. We know that $E[X]=e^{\mu+\frac{\sigma^2}{2}}$, $E[X^...
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0answers
24 views

How would I use a normal distribution approximation to perform a hypothesis test? [closed]

A company produces wine glasses in large quantities. It is known from previous records that 1% of the glasses have a hairline fracture. In a separate experiment, a random sample of 300 glasses were ...
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1answer
14 views

$D=\{ (x,y):0\leq y\leq x\leq 1\}$find $f_{X,Y}(x,y)$ and $P(X\leq a, Y\leq b)$

$D=\{ (x,y):0\leq y\leq x\leq 1\}$ I have to find $f_{X,Y}(x,y)$ and $P(X\leq a, Y\leq b)$ I found $f_{X,Y}(x,y)=2$ (this is correct) , but I can't find $P(X\leq a, Y\leq b)$. My solution: $ \...
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0answers
49 views

Suppose Y is a random vairable with $E(|Y|^\alpha)<\infty$ for some $\alpha > 0$, Then $E(|Y|^\beta)<\infty$ for $0\leq \beta \leq \alpha$

During my studies for intro. to probability (undergrad) we learned about Jensen's inequality. The lecturer showed as the following theorem during one of our lectures: Suppose $Y$ is a random variable ...
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1answer
27 views

Sampling distribution problem

Students height is a normal distribution with $mean=167cm$ and $standard~deviation=3cm$. If we choose 100 students independently, what is the probability that at least 55 of them have height less than ...
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1answer
39 views

Relation between $P(X>x,Y<y)$ and $P(X<x,Y<y)$ [closed]

Is there a simple relation between $P(X>x,Y<y)$ and $P(X<x,Y<y)$. Just like $P(Z<z)=1-P(Z>z)$.
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93 views

100 sided dice continue with cost

You are given a die with 100 sides. One side has 1 dot, one has 2 dots and so on up until 100. You are given a chance to roll the die and, however many dots come up, you can choose to (a): take that ...
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0answers
11 views

Inference On a Selectively Revealed Sample

I think this question may be related to cryptography, so I may have the wrong stack exchange, but I am not really sure. Suppose there are two people Sam and Pam. Suppose we have a distribution, a set ...
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1answer
31 views

Find the distribution of X.

Let X be a random variable that assumes all the non-negative integral values. We are given that for any positive integer: $k · \mathbb{P}(X = k) = 10 · \mathbb{P}(X = k − 1)$ Find the distribution of ...
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1answer
25 views

Deriving uniformly most powerful test for constant and non-constant density

Let's consider $\Theta = \{0, 1\}$ and $X$ random variable with density $f(x;0) = 1$ and $f(x; 1) = 3x^2$ for $x \in [0, 1]$. I want to find the uniformly powerful test of size $\alpha = 0.95$ for $...
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1answer
28 views

Bernoulli experiment - hypothesis testing with specific scenario

Let's consider null hypothesis that we want to examine, that probability of success is smaller than $\frac 1 2$ in Bernouli distribution. We also have independent sample of $20$ observations: $x_1, ...
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1answer
69 views

I don't understand the approach the solution manual is using towards answering part(b) [duplicate]

A test for the presence of a certain disease has probability .20 of giving a false-positive reading (indicating that an individual has the disease when this is not the case) and probability .10 of ...
2
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2answers
79 views

Fixed-rate sampling without replacement

Suppose that we have a population of $N$ elements $E=\{e_1,e_2,...,e_N\}$, and a corresponding set of desired sampling probabilities $P=\{p_1,p_2,...,p_N\}$. Each element $e_i\in E$ should be sampled ...
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0answers
68 views

How can you measure how "shuffled" a deck of cards is?

A few days ago I asked for some methods of measuring how shuffled a deck of cards was. Predictably there were a lot of suggested methods, which got me thinking, which is the best one? I think it'd be ...
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1answer
38 views

Expected number of days it will take for two seeds to grow if each seed has a 50% chance to grow each day

I came across this video by Presh Talkwaker: https://www.youtube.com/watch?v=RY7YKSw1t_M. I attempted to solve the problem by first figuring out the expected number of days for a single seed. Per the ...
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1answer
36 views

What is the probability of winning a bet on 30 of the 37 pockets at a roulette table?

At a roulette table, there is a wheel containing 37 possible pockets a ball can land on (each are marked 0, 1, 2, 3, etc. through to 36) and it is equally likely the ball could land on any of these ...
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0answers
21 views

In how many ways can a math class (10 students and 1 teacher) line up if the tallest and shortest people are not next to each other? [closed]

In how many ways can a math class (10 students and 1 teacher) line up if the (tallest and shortest people are not next to each other)
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0answers
33 views

Is sub-Gaussian random variable invariant to translation?

Say random variable X is sub-Gaussian and $E(X)\neq 0$. Is $X-E(X)$ still a sub-Gaussian random variable? How to verify it theoretically?
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2answers
40 views

Why is the second derivative of the probability distribution function in this case not valid probability density function?

I am asked to find the covariance of $X$ and $Z=min(X,Y)$, where $X$ has exponential distribution $\epsilon(2)$, $Y$ has exponential distribution $\epsilon(3)$ and $X$ and $Y$ are independent. I am ...
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0answers
19 views

One example about modification and indistinguishable

Let $U$ be a r.v. which is uniformly distributed on the interval $[0,1]$. The probability space on which $U$ is defined is denoted by $(\Omega, F, P)$. Define two stochastic processes $\{X_t: t\in [0,...
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4answers
89 views

Does $E(Y)=0$ hold in this setting?

Given two continuous random variables $X$ and $Y$ with the same support, we have $E(X)=0$ and $Y\leq|X|$. Does this setting imply $E(Y)=0$? My proof: $Y\leq|X|\Longrightarrow -X\leq Y\leq X \...
3
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1answer
58 views

Expected number of samples to estimate the mode of categorical distribution

Suppose I have a categorical distribution with pmf $(p_1, \dots, p_n)$. What is the expected number of iid samples $\mathcal{S} = \{x_1, \dots, x_m\}$ I have to samples for the empirical mode to be ...
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0answers
20 views

Cross Entropy and Log Loss Relationship

I'm curious about the relationship between the Cross-Entropy function and the sum-log-loss function when doing multi-class logistic regression applied to neural networks. A setup would be as follows: ...
2
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1answer
41 views

Adding or multiplying probability

In spite of the titles, this is not a duplicate of Multiplying or adding Probabilities If an inefficient mask is 38% protection against viral transmission, then it gives a 62% probability of passing ...
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2answers
56 views

Probability of picking a number from a set with a uncountably infinite number of positive numbers and a countably infinite number of negative numbers?

Sorry if this is a stupid question. I couldn't see anything else quite the same as this, although I found some similar questions. Please remove this if it is a duplicate. (I hardly understand sets so ...
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1answer
58 views

Knock out tournament question (asked before on MSE)... not getting the correct answer

Knock out tournament 1 If it is given that $P_1$ wins in the third round.Find the probability that $P_2$ loses in the second round. 8n players $P_1$, $P_2$, $P_3$, .....$P{_8}{_n}$ play a knock out ...
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2answers
75 views

Given that $k$ of the $n$ balls are blue, what is the conditional probability that the first ball chosen is blue?

A total of n balls are sequentially and randomly chosen, without replacement, from an urn containing $r$ red and $b$ blue balls. Given that $k$ of the $n$ balls are blue, what is the conditional ...
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1answer
51 views

Roll 4 dice. What is the probability that any 2 of them will have a sum of 7? [duplicate]

Given a set of 4 dice, what is the probability on a given roll that any 2 of them will have a sum of 7? (Assume that the two dice are selected in favor of getting a $7$, such that if it's possible to ...
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0answers
46 views

cube probability [closed]

I have one cube which has 4 different painted colours on each of its 6 sides. How many rotational combinations do I have.? If I now have 8 cubes each with 4 different colours from the same subset of ...
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1answer
45 views

Calculating return to player in a slot machine, where a temporary bonus game raises the prizes

I am trying to program a machine that has 3 independent "reels", that are "rolled", and stops on 3 random icons The base machine is a basic slot with e.g. 100 possible combinations....
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0answers
23 views

Set of sets balanced under random subset selection?

Consider the construction of a set $S$ of $N$ sets $s$ $$S=\{s_1,s_2,...,s_N\},$$ where each set $s_i$ consists of $k$ distinct numbers $q\in\{1,2,...,n\}$ $$s_i=\{q_{i,1},q_{i,2},...,q_{i,k}\}.$$ (...
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1answer
65 views

What is the probability of getting exactly two pairs in a poker hand of $7$ cards?

Problem: What is the probability of getting exactly two pairs in a poker hand of $7$ cards? Note: Only $5$ cards count, and the cards that count will be determined by the player holding the cards. ...
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0answers
49 views

Maximal value of an expectation

I've encountered the following problem recently: Random variables $X$ and $Y$ are normal, $EX = EY = 0$, $Var(X) = 20$, $Var(Y) = \frac15$. Given that the correlation coefficient $\rho_{X, Y} = - \...
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1answer
29 views

Simplifying expression involving the multinomial distribution

I'm interested in simplifying the expression $$ \sum_{\substack{x_1+\ldots+x_N=M \\ x_1,\ldots,x_N\ge 1}} \frac{1}{N^M} \frac{M!}{x_1!...x_N!}, \quad M\ge N $$ which corresponds to the probability ...
2
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1answer
34 views

Joint cdf and conditional expectation problem

Suppose $X, Y$ are continuous RVs with joint pdf $f(x, y) = 0.5$ for $0\leq x\leq y\leq 2$ and $f(x, y) = 0$ otherwise. (i) Find the cdf of $Y$. (ii) Compute $P(X < 0.5 | Y = 1.5)$. Are $X$ and $Y$ ...