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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6 views

Poisson Process with continuous rate, Finding Conditional Number of Arrivals

Poisson with customer arrival to the shop rate given by $\lambda (t)=16-(t-4)^2$ Calculate $P(N(5)-N(3)=40|N(4)=70)$ where $N(i)$ means the number of arrivals in the first $i$ hours. The shop ...
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0answers
7 views

Probability of a vector of normal distribution

Given the set of vectors $\{\mathbf{g}^{1}, \ldots, \mathbf{g}^{N-1} \}$ where $\mathbf{g}^{i} \in \mathbf{R}^M$. Assume that $N \leq M$ and elements of $\mathbf{g}^{i}$ follows normal distribution, ...
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2answers
26 views

Find the distribution of $Y = -\log (1-X)$ given that $X\sim U(0,1)$.

If $X \sim U (0,1)$ then if we define a new random variable $Y=-\log (1-X)$ then what will be distribution of $Y$. Please explain.
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0answers
18 views

Appropriate distribution for set of probabilities $p_1 ,…, p_n$

I am doing some evaluation of a system, that has set of probabilities $p_i$ $i= \in \{1,...,N\}$, I need to model them as random variables such that : $$ \sum_i p_i \leq 1$$ and $$ 0 \leq p_i \leq 1 ...
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0answers
10 views

Showing a relation between binomial and negative binomial analytically

If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$. While it is possible to show that using the definition of binomial and ...
1
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1answer
16 views

Derive the distribution of $Z$ given two identically and independently exponentially r.v.s?

$$Z=\frac{X}{X+Y}$$ $(X,Y)$ are iid r.v.s with $$f(x)=\lambda e^{-\lambda x}$$ We are asked to condition on $Y$ to derive the distribution of $Z$; $F(t)$ and $f_Z(z)$. I don't know where to ...
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1answer
19 views

What is $\operatorname{Pr}\{X_j=0|X_i=k\}$

Suppose $u_n=\operatorname{Pr}\{X_n=0|X_0=1\}$ What is $\operatorname{Pr}\{X_j=0|X_i=k\}$, where $\{X_n\}$ is a branching process and $k\geq 0$, if we were to write the answer in terms of the ...
3
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0answers
16 views

Markov Chain: Steady State Distribution.

A total of $M$ balls are divided between two urns A and B. A ball is chosen uniformly at random. If it is chosen from urn A then it is placed in urn B with probability $b$ and otherwise it is returned ...
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0answers
3 views

If $P$ is a transition matrix, and $m_{ij}$ is the mean return time, how can I show that $m_ij = 1+ \sum_{k \neq j}P_{ik}m_{kj}$?

If $P$ is a transition probability matrix of a finite state regular Markov Chain, and $m_{ij}$ is the mean return time, how can I show that $m_{ij} = 1+ \sum_{k \neq j}P_{ik}m_{kj}$? It seems rather ...
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0answers
16 views

show that Y1 is unbiased for θ and find its variance

let x1,....xn IID exponential (θ) Y1={x1+3x2+5x5}/9 Y2= Sigma Xi 1- show that Y1 is unbiased for θ and find its variance. 2-show that Y2 is sufficient for θ 3- let u = E[Y1/Y2] compute u ...
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1answer
21 views

Expected value problem: flip $6$ fair coins before we obtain $3$ heads and $3$ tails?

How many times on average (expected value) must we flip $6$ fair coins before we obtain $3$ heads and $3$ tails? I know I need $∑ xp(x)$. I just don't know how to apply it.
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1answer
19 views

Why does $E(C\cdot \epsilon\; \vert\; C\cdot X) = E(C\cdot \epsilon\; \vert\; X)$?

Let $C$ be an $n\times n$ matrix, $X$ is $n \times k$, $\epsilon$ is $n \times 1$ This is taken from a simply proof of strict exogeneity in an Econometrics textbook by Hayashi. The explanation he ...
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0answers
6 views

When does a periodic but positive recurrent markov chain have a limiting distribution

So I know it's a fact that an aperiodic, finite state, irreducible (so positive recurrent) markov chain has a unique stationary distribution which is limiting. However, I am curious if there is a ...
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2answers
31 views

What is the probability that a psychic correctly “predicts” the outcome of at least 5 out of 10 coin flips?

Assume the psychic is actually just randomly guessing on each flip. The attempt: let E be the event in question number of outcomes per flip = 2 chance of correctly guessing the correct outcome = ...
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1answer
21 views

Troubles With The Beginning

The following is the question I'm having a bit of troubles starting: Musicnotes.com sells sheet music in the following genres: rock jazz, new age, and country. An experiment consists of recording the ...
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0answers
14 views

How to Calculate Covariance of Branching Process

Suppose we have a branching process $\{X_n\}$, where $X_0=1$. How would I go about calculating the covariance $\operatorname{Cov}(X_j,X_i)$ for $i\leq j$? Not sure how to start, so hint would nice to ...
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1answer
17 views

Prove a conditional distribution is uniformly distributed across a given interval?

$X$ and $Y$ are independent random variables identically exponentially distributed with $\lambda$. Take $Z=X+Y$. Show that $(X|Z=z)$ is uniformly distributed over $(0<x<z)$. Then, find ...
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0answers
18 views

How to determine the limiting distribution of a Markov Chain which can only increment up or down a state at every stage?

I have a random walk Markov chain that has states from $0$ to $N$. The conditions are that when the chain is at $0$, the chain will go to state $1$ with probability $1$. When the chain is at state ...
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0answers
51 views

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? [on hold]

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? How should I approach this problem? Okay I think it's C(10, 2) because I already have ...
2
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1answer
29 views

Finding expression for probability given its PGF

Consider the probability generating function for a random variable $X$: $\varphi_X(s)=\dfrac{7-3s}{15-14s+3s^2}$ Find an expression for $P(X=k)$, for $k\in\mathbb{N}$ My attempt was to break ...
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1answer
12 views

Doubt with Notation on Conditional Expected Value Demonstration

I´m having trouble writing a demonstration for the Conditional Expected Value using $\sigma$-algebra. I know its really simple and actually logic but I just can´t find the way to write it. Hope anyone ...
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2answers
26 views

Probability of a fair sequence of tosses ending on two successive tails given the first toss was a head?

Suppose a coin is tossed repeatedly until either two successive heads appear or two successive tails appear. Then, assume that the first coin toss results in a head. I would like to find the ...
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0answers
27 views

Probability Random Variables Fall in an Interval

I've been trying to figure out a counting problem and can't wrap my head around how to calculate the probability. If we let $X_{1}, . . . , X_{10}$ be independent random variable with a uniform(0,1) ...
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0answers
20 views

Mean distance of random points on a rectangular grid

I have a $N\times N$ grid of side $L$. Each gridpoint can be black or white and a ratio $r$ of the points is black. I want to predict the mean distance between two black points. The most appropriate ...
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0answers
9 views

Failure boundary for simple routing problem

As an absolute beginner concerning probability theory I am currently trying to solve the following problem: Given a grid that has $x$ columns (here $x = 4$) and $y$ rows (here $y = 5$), we insert a ...
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0answers
13 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
4
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2answers
115 views

r distinct balls in N boxes

If r distinct balls are distributed at random into N (N ≤ r) boxes, what is the probability that box 1 will receive exactly j balls ( 0 ≤ 𝒋 ≤ r)? my solution is [sample space] =$ N^r $ ...
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5answers
33 views

Probabilty derivation using axioms

$$P((A \cap B^c) \cup (A^c \cap B))=P(A) + P(B) -2P(A \cap B).$$ I need to show this holds. I see it with Venn diagrams but I need to show it using only the axiom, for the union of two disjoint sets: ...
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1answer
25 views

Conditional Probability

An incoming freshman Mark believes that he has a 25% chance of earning a GPA in the 3.5 to 4 range, a 35% chance of graduating with a 3.0 to 3.5 GPA and 40% chance of finishing with a GPA less than ...
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0answers
20 views

Obtaining a percentage from a range [on hold]

In statistics, having an $[a,b]$ range how can I obtain the percentage of the distribution that is included in it? Thanks.
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1answer
8 views

Can I calculate the probality of a test being true $Pr(T)$ from $Pr(T|V)+ Pr(T|not V)$ if I know that Pr(V)+Pr(not V)=1?

If have $T$="The virus test is positive." and $V$="There really is a virus." and I know that $Pr(V)+Pr(\bar{V})=1$, can I then say that $Pr(T)=Pr(T|V)+Pr(T|\bar{V})$ and how do I show that ...
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2answers
33 views

How to compute a “luck percentile” from a set of random numbers or die rolls

I think it's easiest if I start with my actual use-case: In a video game (XCOM), soldiers shoot at aliens. When they do, they have a % chance to hit. Hitting deals damage. I want to look at each ...
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2answers
13 views

Survival bias and probability

Imagine the following situation: A new virus is discovered that is believed to have infected 20% of the population. Anyone infected with the virus has a chance of 50% of dying in their sleep every ...
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0answers
15 views

renewal process and probability(exercice) [on hold]

Let $N_t$ a renewal process. Let $A_t=t-S_{N_t-1}$, $S_{N_t}=X_1+...+X_{N_t}$ with $X_i$ the jumps moments. Let $Z_A(t)=P(A_t \leq u)$ 1) How to show $P(A_t \leq u |X_1=x)=P(A_t \leq u |X_1 \geq t)$ ...
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1answer
25 views

Marginal distribution from a Poisson distribution where intensity is exponentially distributed?

Given that $N$ is Poisson distributed with a random intensity $Y$, the conditional distribution of $(X|Y)$ is defined as, for $n=0,1,\dots$ $$P[N=n|Y=\lambda]=e^{-\lambda}\lambda^n\frac{1}{n!}$$ $Y$ ...
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2answers
20 views

Probability same outcome 3 times in a row.

I am doing some old exam questions - and I don't know the answer, can some one calculate the result and show how you did it?
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1answer
24 views

How do you picture: $\Pr(B|A)$ shrunk down by $\Pr(A)$?

I do not understand how to picture and visualise the following explanation: $\color{green}{[P1.]}$ Suppose you were to grab the edges of $A$ and stretch it out so it covers all of $\Omega$. $B$ ...
4
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1answer
21 views

Ehrenfest Chain: stationary distribution

In the Ehrenfest Chain model: There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn. Let $X_n$ be the # of ...
2
votes
1answer
14 views

Probability matrices in an online game or how to approach matching players to maps to achieve better user experience

Probably I had nonstandart question, but I hope to find some help and valueable advice. Assume I have an online game with $n$ players (let's say $n$ is about 100.000). There's also $m$ maps ($m$ is ...
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0answers
14 views

Probabilities of each waitlist person

Coming from this, $10$ Applicants for a exclusive club membership. I found that you can use total probability to consider existing and old members leaving/returning as members in the club, ended up ...
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1answer
16 views

Expected Number of Draws without Replacement

An urn contains a white balls, b blue balls and c red balls. Balls are drawn one by one without replacement. Find the expected number of draws needed to get all 3 colors.
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1answer
22 views

Difference between $F_X(x)$ and $F(x)$ in probability?

What is this difference in notation between $F_X(x)$ and $F(x)$? (where $F(x) = P\{X \leq x\}$ Thanks.
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0answers
8 views

Expected values of Hermite polynomials under Gaussian distribution

On Wikipedia there's a nice result stating that $$E[He_n(X)]=\mu^n,$$ where $He_n$ is the (probabilists') Hermite polynomial of order $n$ and $X$ is a $N(\mu, 1)$ random variable. I'm interested in ...
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2answers
17 views

Find PDF on $[0,6]$ such that $P([1,3]) = 0.5$

Find a probability density function $f$ on $[0,6] \subset \mathbb{R}$, such that $\mathbb{P}([1,3]) = 0.5$ That is we need to find an $f$, such that $\int_{[0,6]} f(x)dx = 1$ and $\int_{1}^{3} ...
0
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0answers
10 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
2
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0answers
10 views

Weak convergence and convergence of moments

Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ such that $X:\Omega\rightarrow \mathbb{R}$. Suppose that $X\sim N(\mu, \sigma^2)$. Consider a random ...
2
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1answer
23 views

How many ways can we deal a 13-card hand with at least one suit that does not appear?

In dealing a $13$-hand card that with at least $1$ suit that does not appear, I came up with this: We can choose $3$ of the $4$ suits, as in $3 \choose 4$, and then $13$ cards out of the $39$ cards ...
2
votes
1answer
41 views

Probability of winning a game between players A and B?

The following problem is from A First Course in Probability by Sheldon Ross, and it was assigned as homework by my professor. I was wondering if you guys could help me find a answer to the problem. ...
0
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2answers
49 views

Factory with $3$ production lines [on hold]

A factory has $3$ production lines $A, B$ and $C$ contributing $20$%, $30$% and $50$%, respectively, to its total output. The percentages of substandard items produced by lines $A, B$ and $C$ are $10, ...
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0answers
10 views

Cdf of truncated distribution

Let $X$ be a random variable with density $f_x$ and distribution function $F_x$. Define the interval $I = (a,b)$. Given that we know these and the inverse distribution function $F^{-1}_x$, how can we ...