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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1answer
33 views

Quick question about equality of random variables

I have two RV $X$ and $Y$, and I am asked to find the probability of $P(X=Y)$. To tackle to problem, I've conditioned on $X$ and used the law of total probability to write : $ P(X = Y) = \...
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1answer
41 views

Cumulative Multinomial Distribution does not seem to add up to 1.00 for parameters >3

For a multinomial distribution $P = ( n! / (\prod_{i=1}^k {n_i!}) ) * ( \prod_{i=1}^k {{p_i}^{n_i}})$ If you sum the resulting multinomial distribution for every possible unique frequency for k=2 for ...
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0answers
21 views

Determine Total PDF of Conditional PDFs

Suppose I have a scaled probability density, defined as a function of a random variable $t$, where: $f^{'}_{j}(t,j) = P_{j}(t,j)f_{j}(t, j)$ where $f^{'}_{j}(t, j)$ is an aribtrary pdf, defined by ...
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2answers
76 views

Why is $P(|x|\ge\varepsilon) = P(x^2\ge \varepsilon^2)$?

I am trying to understand the Chebyshev inequality. The step I cannot really follow is the identity: $P(|x-\mu|\ge\varepsilon) = P((x-\mu)^2 \ge \varepsilon^2)$. Can anyone explain the intuition ...
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0answers
46 views

Optimal Solution to a Game With Two Sequences of Bits [duplicate]

I was given a nice riddle, and solved it, and I can't tell if my solution is optimal or not. The riddle is a follows: Two infinite sequences $a_n, b_n$ of bits are randomized. The sequence $a_n$ is ...
2
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1answer
46 views

Intitutive explanation of why probability of getting an even integer is more when random numbers are multiplied.

If n(>1) random integers are selected, then the probability of their product being odd is $1/2^n$, which is less than that of the product being even. But this doesn't sound intuitive to me. If ...
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0answers
16 views

Random walk with absorbing barriers: total probability of absorpion

Consider a random walk that moves up by 1 unit with probability $p$ and down by one unit with probability $1-p$. It starts at $0$ and there is one absorbing bareer at $-2$. How can I calculate the ...
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0answers
12 views

Provide a detailed explanation of why there is no space of probability that is both a Laplace and a geometric model of probability.

Could you help me answer one question? The question is: Provide a detailed explanation of why there is no space of probability that is both a Laplace and a geometric model of probability. I tried to ...
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0answers
41 views

Confusion regarding probability problem involving 3 archers

In a recent post , the following problem was asked. There was a solution provided; however, I was unable to fully comprehend it. I have copied verbatim the part that I was confused about here - ...
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1answer
38 views

Evaluate $tr(ADAD)$ where $D$ is a diagonal matrix with iid entries from $N(0,1)$ on diagonal and $A$ is a deterministic $n \times n$ matrix

Let $D$ be a random $n \times n$ matrix with iid entries from $N(0,1)$ on the diagonal and let $A$ be a symmetric deterministic $n \times n$ matrix. Question. What does $tr(ADAD)$ evaluate to ?
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1answer
39 views

Is a uniformly random $r$-regular bipartite graph $r$-edge connected with high probability?

A graph is $r$-edge connected if the number of edges in a minimum cut is at least $r$. It is known that a random $r$-regular graph is $r$- vertex connected (which implies $r$-edge connected) with high ...
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1answer
39 views

Why do we approximate sample stdev using sample stdev / sqrt(n) when population stdev is unknown? [closed]

In this stackoverflow stdev estimation question, we are discussing about when to use z-distribution and t-distribution. Now, I'm getting confused here why are we taking the indirect & apparently ...
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1answer
42 views

What is the probability that a specific group of four students will be chosen out of 36 students?

Nine groups of four are formed from a group of 36 students. What is the probability that a particular group of four students will be chosen? Order is unimportant. My first approach was to calculate ...
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1answer
49 views

Correlation bound proof

Disclaimer: I am not talking about the correlation coefficient $\rho \in [-1,1]$ The correlation, according to a textbook I have, is defined as $m_{11} = E[XY]$ given the Model $(X,Y,f(x,y))$ of $2$ ...
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1answer
30 views

Likelihood ratio test for Bernoulli distributions

Let's say that we have sample $x_1, ..., x_n \sim \textrm{Bernouli}(\theta)$ and consider hypothesis: $H_0: \theta_0 = \theta$ and $H_1: \theta_0 \neq \theta$. I want to find form of likelihood ratio ...
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0answers
20 views

Simple Probability question from observing a graph out of Henk Tijims- Understanding Probability book

Good evening, The graph shown below is in reference to the text as follows: The highlighted text below in bold is what I don't understand. How was this deduced from the graph. Any help is greatly ...
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0answers
17 views

Conditional expectation-cond.b.t.square [closed]

Can someone solve this task? task
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0answers
31 views

Laplace probability estimate

I am trying to calculate the Laplace probability estimate according to this formula: The correct answer is $0.733$ but I just don't know how to find the number of examples that belong to the Class ...
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0answers
35 views

Is my reasoning correct for this (probably) markov chain problem?

Here the problem : In the morning, Alice goes from her home to her office and returns home in the evening. She has a scarf : on each journey, if it is cold and her scarf is in place (in her office ...
2
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1answer
28 views

Let $N(t)$ be a Poisson process. Calculate $P(N(s)\neq N(t))$ and $P(N(s)=0, N(t)=1)$

Let $N(t)$ be a Poisson process with parameter $\lambda$ . Calculate a) $P(N(s)=0, N(t)=1)$ b) $P(N(s)\neq N(t))$ My try: Let $0\leq s < t$ a) $P(N(s)=0, N(t)=1) = P(N(s)=0, N(t)-N(s)=1)=P(N(s)=0)(...
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0answers
17 views

What is the best way to approximate Discrete Markov Random Field? [closed]

Approximating Discrete Markov Random Field (MRF) is also called Discrete Markov Random Field Relaxation in literature. A 3 values 4 variables MRF can be defined as $$\sum_{a_1=0}^{2}\sum_{a_2=0}^{2}\...
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1answer
21 views

Extracting $P_0$ from generating function

I have the following generating function for stochastic process $$\sum_n z^n p_n=\left[1-\frac{(1-z)}{\Lambda(t)}\right]^{n_0},$$ and I want to extract the probability $p_0(t)$ but I am confused how I ...
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1answer
51 views

How do I calculate an expected value?

I read the introduction, the problem, couldn't solve it, viewed the explanation, but still don't understand it. You calculate an expected value by adding up weighed averages, don't you (at least, ...
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0answers
43 views

Jensen's inequality to prove moment exists?

let $X$ be a random variable, and let $\alpha , \beta$ 2 scalars in $Z$ such that: $0<\alpha <\beta$. Prove that if the $\beta$ moment for $|X|$ exists then the $\alpha$ moment for $|X|$ exists ...
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1answer
54 views

About a problem on Chebyshev's inequality

Regarding a solved problem from one Book from Sheldon Ross as below: Problem: Suppose that it is known that the number of items produced in a factory during a week is a random variable with mean 50 b)...
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2answers
121 views

Expected number of rolls until all dice are removed

There are $n$ fair dice. They are all tossed every time except the dice that are removed. A dice is removed if $3$ is rolled. What is the expected number of rolls? Any help would be appreciated. My ...
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0answers
31 views

Random Variable can take no values at all?

Is it possible for discrete random variable not to have any value? ie $P_X(x)=0$? I think it's not possible because we should get that: $\sum_xP_X(x) =1$. Am I right?
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0answers
21 views

Minimal reward for guessing the correct passcode

There is a 4-character passcode consisting of letters A,B and C. You try to guess the passcode. First wrong guess costs you 1 dollar , second wrong guess costs you 3 dollars, third 5 dollars,.... What ...
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1answer
46 views

Why does Polya's random walk theorem for $d>3$ follow from $d=3$?

Im reading this paper in the Direct counting argument section(I'm only interested in this method of proof) and it is said that once we have proven $d=3$, the $d>3$ case follows as a generalisation ...
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1answer
34 views

Getting P(A, C | B) from P(B | A, C) using Bayes' [closed]

I found online that P(B|A, C) = P(B, A, C) / P(A, C). How can I calculate P(A, C | B) using Bayes'?
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0answers
38 views

Renewal Process Identity

Setup: Consider a renewal process where $Y_i\overset{iid}{\sim} p$ for $i\in \mathbb{Z}_{+}$ and $p$ is a probability distribution $$p:\mathbb{Z}_{+}\rightarrow [0,1].$$ If we define $Z_k=\sum_{i=0}^{...
2
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1answer
55 views

How to use Markov inequality to solve this problem

Markov inequality: For non-negative random variable $X$ and real number $v$, then $Pr[X \geq v] \leq \frac{E(X)}{v}$ My question is: Let $0<a,b<1$, and Let $Y$ be a random variable ranging in ...
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0answers
20 views

Expectation of linear combination of randomly chosen elements of a vector whose expectation is zero

I asked this question to address the following question: Let $\mathbf{b} \in \mathbb{R}^n$ be a random vector distributed by some distribution $\mathcal{D}$, i.e., $\mathbf{b} \sim \mathcal{D}$. ...
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1answer
56 views

Different Versions of Jensen's Inequality?

According to wikipedia: Jensen's inequality is ${\displaystyle f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).}$ With this, I can easily prove Inequality of arithmetic and geometric means, in ...
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0answers
23 views

Convergence in L2 and Uniform Boundness [closed]

I was reading about the following algorithm (https://en.wikipedia.org/wiki/Stochastic_approximation#Robbins%E2%80%93Monro_algorithm): I am trying to understand what is meant here by L2 Convergence ...
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1answer
60 views

Equivalent Probability Distributions

I guess the source of the question would be this, which contains a similar problem: Let $U_1$ and $U_2$ be two independent random variables, each with uniform distribution on $[0,1]$. Let the random ...
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1answer
59 views

$X,Y \sim N(\mu,\sigma^2)$ Suppose $\sigma_X^2=\sigma_Y^2 \neq 0 \implies X+Y,X-Y$ are independent.

$X,Y \sim N(\mu,\sigma^2)$ Suppose $\sigma_X^2=\sigma_Y^2 \neq 0 \implies X+Y,X-Y$ are independent. I know that $\sigma_{X+Y}^2=\sigma_X^2+\sigma_Y^2 , \sigma_{X-Y}^2=\sigma_X^2+\sigma_Y^2$. If I'll ...
2
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2answers
63 views

The probability of mistake of randomized program.

Consider this program to detect odd and even numbers: If the input number is even, it prints $even$. If the number is odd, it has a $4/5$ probability of printing $odd$ and $1/5$ probability of ...
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1answer
31 views

Probability and binomials choosing phones

So assume we have 100 phones of some sort and that we choose 15 of them. We know that on average 6 of them are faulty and 94 are fine. What is the probability of us choosing 15 phones from the 100 so ...
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0answers
19 views

When do we use p-value or confidence intervals

Let's take a simple example, where we are given the daily average costs of a student 1.2 0.8 0.6 1.1 1.2 0.9 1.5 0.9 1.0. And we have to determine whether we can reject the hypothesis $H_0$: the mean ...
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3answers
106 views

Meaning of $\frac{P(X\cap Y)}{P(X)P(Y)}$

Imagine that we have a set $\Omega$ and $X$ and $Y$ are events that can happen, I mean, $P(X),P(Y)>0$. Then, what does it mean the ratio $\frac{P(X\cap Y)}{P(X)P(Y)}$? I know that $\frac{P(X\cap Y)}...
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0answers
32 views

Multiplication of two multivariate probability density functions (pdfs) of different variables

Greetings first of all I'm not that well versed in probability theory, my appologies. Regardless, I want to use probability theory for solving an data association problem. Basically I've got the law ...
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1answer
44 views

Find the mean of the value of the coins.

A bag contains a large number of coins. Half of them are \$1 coins, one-third are \$2 coins, and the remainder are $5 coins. Find the mean of the value of the coins. If a random sample of 2 coins is ...
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1answer
44 views

Is there a shortcut to solving this probability problem?

Assume that person A throws a dice and gets 1 point every time they get a six, this is repeated 12 times. Now we have person B that also throws dice and gets 1 point every time they get a five, this ...
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1answer
40 views

Players $A, B$ are gamblers, B rolls the cube. A is a cheater, he rolled his cube again and again till he get an equal or higher number than $B$.

Players $A, B$ are gamblers, B rolls the cube. A is a cheater, he rolled his cube again and again till he get an equal or higher number than $B$. Given that $A$ rolled his cube twice., find the ...
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0answers
19 views

Identifying the pdf of a posterior predictive distribution

Basically, after integrating over my exponential and gamma function I end up with the pdf of the posterior predictive distribution: $$p(y_f\vert y) = \frac{\Gamma(\tilde{\alpha}+1)}{(y_f +\tilde{\beta}...
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2answers
41 views

Find the probability that more heads occur than tails. [closed]

A biased coin with the probability of getting head = 1/3, is tossed 3 times. Find the probability that more heads occur than tails. Find the probability that all the outcomes are the same. Find the ...
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1answer
51 views

Prove Jensen's inequality?

I was interested to see a proof for Jensen's inequality for the following variant: Let $X$ be a discrete random variable with finite expected value and let $h:\mathbb{R} \to \mathbb{R}$ be a convex ...
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1answer
22 views

How would you calculate expected value of sales leads?

Simplified situation: I have 1000 leads, but can only pursue 100 of them due to limited resources (salespeople). Each of these leads can be pitched up to N products, each with its own dollar value ...
3
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0answers
35 views

Find the probability that he does not pass either examination?

A student sits for two exams, theory and practical. He estimates that the probability of passing the theory examination (Te) is 0.5, the probability of passing the practical examination (Pa) is 0.3 ...