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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. 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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How do you determine if a pair of random variables are independent?

If $A$ is a subset of $R$ and $X$ is a random variable. I have two variables $X_1$ and $X_2$. $I$ being $1$ if $X$ in subset $A$ and $0$ if not in $A$. Let $U$~$U(0;1)$ and determine if this pair ...
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Why is $ P(−y ≤ X ≤ y) = 2P(0 ≤ X ≤ y)$ and not just $P(0 ≤ X ≤ y)$ when the range of $y$ is $[0, ∞)$

Given that $Y=|X|$ , the range of $y$ is $[0, ∞)$, $P(Y ≤ y) = P(−y ≤ X ≤ y) = 2P(0 ≤ X ≤ y)$ My question is, why does $ P(−y ≤ X ≤ y)$ equal to $2P(0 ≤ X ≤ y)$ and not just $P(0 ≤ X ≤ y)$ ? ...
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2answers
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Having a problem on a paradoxical answer

Suppose that we have PN objects (disks for an example) and we have P slots (or boxes),how many ways can we distribute those PN objects on those P slots so that each slot has exactly N object? i saw a ...
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1answer
18 views

A randomly selected number between 1 and 5

How come $f_X (x)$ is not equal to 1/5? Because it should be $f_X (x) = 1/x$ so $f_X (x) = 1/5$ because x is defined on [1,5] right?
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Poisson Random Variable Question

A radioactive source emits certain particles with a Poisson distribution. The probability of no particle emissions during an hour of observation is $0.4$. What is the probability that the first ...
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transition semigroup and interpretation of excessive function

Let $\{P_t\}$ be a transition probability kernel. We say $f\in L^2(X)$ is a- excessive if for all $t>0$ and $x\in X$, it holds $e^{-at}P_tf(x)\leq f(x)$. My question: How can I interpret a-...
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Probability after a specific sequence of events?

I am working on some probability problems (not homework, I'm actually a 24 year old software developer who is doing some number crunching for a personal project) and wondering if someone who ...
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1answer
31 views

Probability of getting from one point to another given probability that path is open

The points Woodstock and Tunbridge (W and T) are connected above in 3 different scenarios. p and q are the probability that the path is open. The question is what is the probability one can get from W ...
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27 views

Density Function.

Let X be a random variable having density function as 𝑓(𝑥) = 𝑐 (1 − 𝑥) 𝑥 ;0 < 𝑥 < 1. Find the value of c. After solving this question I got the answer as 6, whereas the answer given in ...
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2answers
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Cutting a 1 metre stick, what is the expected length?

I am unable to understand 2 things in this problem: Why is $f_X (x)$ = 1, should it not be 1/x because every point has the same chance of being cut. Does that mean every point has P=1 of being cut? ...
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1answer
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For game where first player to N points wins, find the distribution of win probability and total number of points between players

Two players, A and B, play a series of points in a game with player A winning each point with probability p and player B winning each point with probability q = 1 - p. The first player to win N ...
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1answer
24 views

Maximum Variance of a Distribution

Let the random variable $X$ have the distribution $\mathbb{P}(X=0)=\mathbb{P}(X=2)$, $\mathbb{P}(X=1)=1-2p$ for $0\leq p \leq 1/2$. For what $p$ is the $\mathrm{Var}(X)$ maximum ?
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1answer
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Why does $P(E) < P(F)$ imply that $E \subseteq F$?

Why does $P(E) < P(F)$ mean that $E \subseteq F$ ? My reasoning (using Venn diagrams): It is seen clearly in the below picture that even if $P(E)<P(F)$, there is still some region in E that is ...
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0answers
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I am having trouble figuring out how many lambda's (births) there are in a given birth-death Markov process problem.

These questions are not for assignment. I am just confused as to how to set up the problem. I also do not need help calculating the problems at hand. I understand that in a birth and death problem, $...
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1answer
21 views

let z1 and z2 be independent standard normal RV, find the pdf of $e^{3Z_1+2Z_2}$

Let Z1 and Z2 be independent standard normal random variables. Find the following The probability density function of $e^{3Z_1+2Z_2}$ The given solutions is as follows: But what doesn't make sense ...
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1answer
16 views

given exponential Cumulative distribution function, finding another Cumulative distribution function with functionl connection

There is given $X$ a random variable with exponential cumulative distribution such that $X~Exp(1)$ so the exponential Cumulative distribution function is: $P(x\le t)= F_x(t)=(1-e^{-t} , 0\le t) \...
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2answers
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Finding $E[X^2]$ for $X \sim Bin(25,0.61)$

I got something rather new and I just wanted to make sure my way of thinking in this field is fine. Suppose $$X\sim Bin(25,0.61)$$ and we are asked to find: $E[X^2]$. So basically I treat this ...
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0answers
19 views

Difference between two standard deviation formulas

I was just wondering about the difference between these two standard deviation formulas? I am extremely confused because with one of the standard deviation formulas, it also requires the standard ...
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1answer
55 views

Cutting a rope randomly and taking the longer piece, cutting the longer piece and taking the shorter piece.

Cut a rope with unit length into two pieces randomly. Cut the longer piece of the first cut into two randomly again. Take the shorter piece from that second cut. What would be the PDF and the expected ...
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1answer
24 views

Bayes Formula for more than 2 events

I have the following result used in one of the paper that I'm reading. I can't seem to get around how it is derived. Please help. $$ P(M\mid D,R) = \dfrac{P(M\mid R)\,P(D\mid M)}{P(M\mid R)\,P(D\mid ...
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1answer
30 views

Probability of a certain 5 card hand from a standard deck

A poker hand consists of 5 cards randomly dealt from a standard deck of cards without replacement. What is the probability that you're dealt a hand that contains exactly one pair of red Queens with ...
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1answer
29 views

How to solve this probability/combinatorics question about balls in bins

Below is the problem that I need to solve Suppose you blindly place five balls labeled A, B, C, D and E inside five bins labeled A, B, C, D and E. What are the chances that, in your selection, no ...
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can you help me understand the inputs for this bayes theorem calculator for my scenario?

I'm trying to calculate the probability of delivering a website based on the conditional delivery of an underlying database. I entered some inputs into this online Bayes Theorem calculator: https://...
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6answers
642 views

Monty Hall Problem-Probability Paradox

I just learned about the Monty Hall Problem and it seemed pretty much amazing to me.I am just a bit confused with it. So,according to the problem we are on a game show, and we are given the choice of ...
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1answer
42 views

Definition of a family of probability measures for Ito diffusions

I have a question concerning the definition of a family of probability measures for the solutions to an Ito diffusion $$X_t^x = x + \int_0^tb(X_s^x)ds + \int_0^t \sigma(X_s^x)dB_s$$ as it's given in ...
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1answer
29 views

Probability of having drawn ace of spades after drawing a card and putting it back in the deck

I have been wondering if the probability of drawing a card from a 52 card deck, obviously 1/52 probability, is different from the following. If you get to draw a card from a 52 card deck and right ...
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1answer
42 views

A game with an $n$-sided die

Suppose you play a game with a fair $n$ sided die (that, if being rolled yields us a discrete random variable uniformly distributed on $\{k \in \mathbb{N}| k \leq n\}$). You play the following ...
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0answers
22 views

A random variable formed by two Normal random variables, under a conditioning process.

Imagine two independent random variables, $X$ $\sim$ $N$ $(\mu_1$,$\sigma_1^2)$ and $Y$ $\sim$ $N$ $(\mu_2$,$\sigma_2^2)$. Now imagine a process whereby one observation of $X$ and one observation of $...
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35 views

Probability/ eggs in boxes

I have n boxes and n eggs, which were hidden randomly in these boxes. The probabilty, that no egg is in the right box, is $$ \sum_{k=2}^n (-1)^k \frac{1}{k!} $$ How can I get to this formula?
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Different ways of computing probability

There are twelve unique watches and five men. Each of the five men are asked to choose a watch for themselves. What is the probability that at-least two of them choose the same watch. I could ...
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1answer
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Joint Exponential Probability Problem

I have a question from A First Course in Probability by Sheldon Ross. Question: Consider two components and three types of shocks. A type 1 shock causes component 1 to fail, a type 2 shock causes ...
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1answer
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How to understand this probability equation?

$\mathbf { x } ( t ) = g ( \mathbf { s } ( t ) ; \xi ) + \mathbf { n } ( t )$, where n(t) denotes the noise or modeling error and ξ the parameters of mapping $g$ How to understand the following ...
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0answers
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Do jumps in Levy processes need to be independent of the process itself?

I have a very basic question about Levy processes. Is the process of the form $$ X_t=\sigma (X_t) B_t + \sum_{i=1}^{N_t}\eta_i(X_t) $$ a Levy process? Here $B_t$ is a standard Brownian motion and $N_t$...
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1answer
28 views

proof that $Y$ follows normal distribution

I'm new to probability and studying multivariate normal distribution. The one thing I don't understand is the linear transformation of multivariate normal distribution.If the $X$ follows Normal ...
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1answer
45 views

Does MLE really care about PDF?

I am wondering, whether MLE really cares whether it operates on proper distributions. Lets take a look at the following situation: likelihood: $$L(\theta \mid x) = \prod_{n}^{N}{f(x_n \mid \theta)}$$ ...
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is this the correct probability equation?

Event A has a probability of 80% Event B has a probability of 90% Event B depends on Event A Solution: (.80 * .90) * .80 = 0.576 probability UPDATE Here are some more details for those who ...
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1answer
31 views

Probability in a comunication network

A communication network is made of nodes conected with wire. The net sends packets in such a way that if one packet is located in an internal node $x$ (internal node is the one connected to more than ...
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0answers
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Independence of sample mean and sample variance

It is well known that under normality assumption, the sample mean and sample variance are independent, by Basu's Theorem. My question is that, is the normal distribution the only distribution whose ...
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1answer
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Probability the event occurs knowing that I received no information

First I want to thank you if you pay attention to my post. I apologize if it seems elementary to you, note that I searched a lot an answer before posting. I'm going to try not to be vague, do not ...
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0answers
35 views

drawing unique elements with replacement

I have a situation where I will draw a random number of balls from an urn with $r$ red balls and $b$ blue balls, with $N=r+b$. The number drawn is $k$, and I know the distribution $k$ comes from. ...
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1answer
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how to calculate cumulative probability for *dependent* events?

The formula for cumulative probability for independent events is easy enough. Just multiply the probability of the events together. For example, 2 independent events, each with a probability of 0....
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0answers
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Expected number of random rows needed to get a full-rank matrix

I have a random $m \times n$ matrix over $F_2$ of full rank (i.e. all columns are independent) where $m$ << $n$. Now, I wish to randomly choose $k$ rows such that the corresponding sub-matrix (...
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1answer
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Example of two non-mutually exclusive events that are dependent?

(1)If two events are independent that implies that they are "non-mutually exclusive". Then by using logic transposition, Non "non-mutually exclusive" events implies they are "not independent" That'...
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If the probability of a dog barking one or more times in a given hour is 84%, then what is the probability of a dog barking in 30 minutes?

Poorly worded title but I don't know what the nature of this probability question is called. I was asked a question: If the probability of a dog barking one or more times in a given hour is 84%, then ...
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Expectation of sum of geometric random variables vs. expectation of Pascal r.v.

Let $\{X_i\}$ be a Bernoulli process, i.e. $X_1, X_2, X_3, \dots$ are i.i.d. Bernoulli variables with parameter $p$. Let $T_k$ be the time at which the $k$th success occurs. I can reason about the ...
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1answer
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What is $\mathbb P(X\in A\mid Y\in B)$?

I'm confuse with formula of conditional probability. In one hand : $$\mathbb P(X\in A,Y\in B)=\int_B \mathbb P(X\in A\mid Y=y)f_Y(y)dy=\int_B\int_A f_{X\mid Y=y}(x)dxf_Y(y)dy.$$ Now, do we have ...
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0answers
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Convergence of Random Variable (convergence in probability)

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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Convergence of random variable ! (Probability) [on hold]

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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1answer
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Simple probability and statistics problem

If there are 12 football teams in a league, how many different bets can you make if you bet on 3 first teams and 2 that will get kicked out of the league? The solutions says its (12*11*10*9*8)/2. ...
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2answers
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Could someone explain explain to what correspond the moments of a random variable?

Could someone explain what are (at least the four first) moments ? (normalized moment to be more precise) Let $X$ a r.v. So the first moment is the expectation. This will correspond to $\mathbb E[X]$...