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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2 views

find the distribution of the random variable $s=\sqrt{\tfrac{R}{1-R}}\cos(\theta)$

I'm stuck with this problem.Can anyone please give me a help? I want to find the distribution of $s$, where R is an uniform random number from $0$ to $1$, and $\theta$ is an uniform random number from ...
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If $(X,Y,Z)$ is jointly normal, what is the distribution of $(X,Y)|Z$?

Let $(X,Y,Z)$ be a jointly normal random vector. I am interested in the distribution of $(X,Y)|Z$. It is normal obviously, but what is its mean and variance matrix in terms of the mean and variance ...
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Does infinite expectation imply regular variation?

Consider a positive random variable $X$ with $\text{E} X = \infty$ (rather than undefined, as with Cauchy RV's). Is this sufficient to conclude $1-F_X(x) = \ell(x) x^{-\alpha-1}$, where $\alpha \in (0,...
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Finding conditional density

$X$ is an exponential random variable with $E[X] = \frac{1}{\lambda}$. $Y$ is an exponential random variable such that $E[Y | X = x] = \frac{1}{x}$. Find $f_{X|Y}(x|y)$. I first find the joint density,...
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Independence of Random Variables A B C [closed]

if A,B,C are independent RV's. How do we show A and A+B+C given A+B are conditionally independent.
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Calculating expected value of infimum of two point distrubution

Let $(X_k)$ be a sequence of random variables as following : $P(X_i=-1)=q$ and $P(X_i=1)=1-q$. I want to calculate $$E[\inf\{n:X_1+X_2+...+X_n>0\}]$$ My work so far Let's denote $Y(x)=\inf\{X_1+X_2+...
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15 views

Distribution function and simulation

Consider the following distribution function $F(x,y)=\max(x+y,1)-1,$ defined on $[0;1]^2.$ Is it possible to find a random variable $X$, such that the distribution function of $X$ is equal to $F$ on $[...
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Proving markov process of squared variable

Let's consider $T=[0,+\infty]$ and let $\tau$ be markov process. I want to check if $\tau^2$ is also markov process i.e. if $\tau^2 \in \mathbb{F}_t$ My work so far $$\{\tau^2 \le t\}=\{\tau \in [-\...
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$P(X_1+X_2<0)=???$

Let $X_1$ and $X_2$ be random variables i.i.d $\mathcal{N}(2,5)$. What is $P(X_1+X_2<0)$? My attempt: X$=X_1+X_2$ has distribution $\mathcal{N}(4,10)$ $P(\textbf{X}<0)=0.3446$ Is it correct?
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Probability of having two cycles of length n/2 in an Erdős-Rényi graph?

Given an Erdős-Rényi graph $\mathbb{G}(n,p)$ (that is, a random graph where each edge exists with probability $p$), what is the probability of having that a graph created according to $\mathbb{G}(n,p)$...
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Determine explicitly the set of reachable $K = {(H · S)_1 : H ∈ H}.$

Let $Ω = {−1, 0, 1},\; \mathbb{P}({−1}) = 1/4,\; \mathbb{P}({0}) = 1/4,\; \mathbb{P}({1}) = 1/2,\; S_0 = 1,\; S_1(ω) = 1+ω$ and $\mathcal{F}$ from $S$ generated filtration. Determine explicitly the ...
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Probability that Sum of Uniform Random Variables Exceeds Value? [closed]

Let $X_i$ be independent and identically distributed with a $U(-1, 1)$ (continuous uniform) distribution. Define $J_n=\sum_{i=1}^nX_i$. Define $P_{n,K}=P(J_n>K)$. Give the distribution (probability ...
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Expectation question mixed with confidence interval [closed]

I encountered this question during a disclosed interview and I had trouble with choosing the method(random walk expectation or Central Limit Theorem) solving this problem. The question is as follow: &...
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Integrating over Conditional Probability

I'm having a little trouble with a probability problem and I can't find anywhere online that answers my question. If I am looking for $P(X = a|Y \leq z)$, is this the same as $\int_{- \infty}^z P(X = ...
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Does it make sense to ask what the probability that a coin is biased (in favour of heads) is?

A coin is tossed $5$ times and $5$ heads are observed. What is the probability the coin is biased in favour of heads? Does this question even have any meaning without using a significance level? Let's ...
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1answer
30 views

Computing Probability given CDF

Given $$F(x) =\begin{cases}(x-1)^2/4,& 1 < x < 3\\0,& x\le 1\\ 1,& x\ge3\end{cases}$$ Is $P(2 < X < 5) = P(2 < X)$ since $F(x) = 1$ when $x\ge 3$? Also would $P(X = 1.5) = f(...
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Set of the probility measuers $W ⊆ L^d$ is convex and closed.

Show that set of the probility measuers $W ⊆ L^d$ is convex and closed, where $L^d$ is a set of all signed measures. I really don´t know how to begin and approach this problem. I would appreciate any ...
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1answer
49 views

Is this random bit generator broken?

Consider a following problem: Suppose, a random bit generator of type $p$ was brought to a random bit generator repair station. Before starting repairing it, the workers from the repair station were ...
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1answer
36 views

Probability puzzle for a fair die

What is the expected number of times we have to roll a fair six-sided die to obtain that the number of times any two faces appear are equal (after the first rolling)?
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Probability of having more heads than another player

Person $A$ and person $B$ have $8$ and $10$ fair coins. They both flipped their coins. What is the probability of person $B$ having more heads than person $A$? (I know how to do the question if number ...
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Joint Distribution of uniform random variable and the sum of this variable with another normal random variable

Given we have two continuous, independent random variables: One is uniformly distributed on the interval [-1,1], so X$\sim$U(-1,1) and the other one is standard normal distributed, so D$\sim\mathcal{N}...
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1answer
31 views

What is the probability of getting red marbles from bag B?

So there are 3 bags: bag A, bag B, and bag C. Bag A contains 2 red marbles and 3 white marbles. Bag B contains 3 red marbles and 5 white marbles. Bag C contains 4 red marbles and 6 white marbles. ...
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1answer
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Show that $M_n = \sum_{i=1}^n \alpha_k(X_k -X_{k-1})$ is a square integrable martingale w.r.t. $\{F_n, \ n\in \mathbb{N}\}$.

$\textbf{question}$ Let $\{F_n, \ n\in \mathbb{N}\}$ be a filtration and $\{X_n, \ n\in \mathbb{N}\}$ a square integrable martingale w.r.t. $\{F_n, \ n\in \mathbb{N}\}$, with $X_0 = 0$ , and such ...
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1answer
11 views

Bayes theorem and probability and statistics

Bag A contains 3 red marbles and 2 green marbles. Bag B contains 5 green marbles only. One marble is randomly drawn from A and put into bag B. Then one marble is randomly drawn from bag B. Find the ...
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28 views

Venn diagram and probability

In a class of 40 students, 22 of them have exempted Mathematics; 7 of them have exempted Computing Studies whereas 15 of them have to study both subjects. a) Show these information in a Venn diagram b)...
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16 views

Joint PMF probability exercise

$X_1$ represents the number of clients in a queue, and $X_2$ the same, but is faster (the queue). (see figure for the pmf) 1 - What's the probability that: a) both queues are empty? b) both queues are ...
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20 views

Probability of randomly selecting a letter that is also contained in a six letter long string

If I had a six-character string of letters and I randomly choose a letter from the alphabet, how would I find the probability that the letter chosen is also in the six-character string? In other words,...
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2answers
30 views

Is force of mortality monotonic function?

I'm trying to do solve some question from the actuarial exam. The task is to map a function with a graph. In the graph the function is increasing on $[0,80]$ and decreasing on $[80,100]$. The options ...
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1answer
26 views

Probability of success on external condition. [closed]

Suppose you have an job interview at company x. You have the liberty of choosing your interview time from 9:30 AM to 4:30 PM. Interview will be for 1 hour. You are a neutral candidate, before the ...
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1answer
23 views

Density of $Y=-\frac{1}{\lambda}\log X$ given that $X$ is a uniform r.v. on $[0,1]$.

As in title given that $X$ is uniform in $[0,1]$ and $Y = -\frac{1}{\lambda}\log X$ ($\lambda \gt 0$), to find the density of $Y$ I'm attempting in the following way: finding the cumulative ...
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Probability that a randomly picked node is in the minimum cut of a graph

I am trying to figure out the probability that a randomly picked node in a graph $G = (V,E)$ belongs to one of the partitions of the minimum cut of $G$, i.e. given a min-cut $(A,B)$, what is the ...
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7 views

Mean and covariance of a normalized normal random variable

Assume that ${\bf x} \in R^{K \times 1}$ is a multivariable normal random variable with mean of $\bf \mu$ and covariance matrix of $\Sigma$. I would like to know the mean and covariance matrix of ${\...
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1answer
15 views

Probability: $ (A'\cap B' \cap C' ) $ and $ (A \setminus (B \cup C )) $

how can I find the probability of the two events: $ P(A)=\frac{1}{2},P(B)=P(C)=\frac{1}{3}, \\ P(A\cap B)=P(B\cap C)=\frac{1}{12}, P(A\cap C)=\frac{1}{9};\\ P(A\cap B\cap C)=\frac{1}{36}$ $ (A'\cap B'...
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Does $p(a | b,c) = p(a|c)$ necessarily imply $p(a|b) = p(a)$?

I'm in a ML course, and we had this math refresher quiz. We were asked to prove (or disprove) the following: $$p(a | b,c) = p(a|c) \to p(a|b) = p(a).$$ It is clear that $a$ is not dependent on $b$, ...
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1answer
35 views

$X_r$ is the number of cycles of length $r$ in a random permutation of $n$ elements.

$X_r$ is the number of cycles of length $r$ in a random permutation of $n$ elements. Find $EX_r, DX_r$. I tried to find $EX_r$ and get it = $\frac{1}{r}$. $DX_r = E(X_r^2) - (EX)^2$. How can i find $E(...
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Complement of conditional independence

This is actually a big question and I'm asking whether my conclusion is right. I have two events A & B and they are independent given an event C. Then we can state this as, As well as, and Then ...
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1answer
39 views

Bayes theorem and Corona virus

I don't know if I understand this correctly. Suppose a test is 99 percent accurate. Now what does that mean exactly, that there is one percent of false negative and one percent false positive? ...
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1answer
33 views

Find the distribution of random variables $τ = \min \{n≥1: X_n> 1\}$ and $X_τ$, calculate $E(τ), E(X_τ)$

There is a sequence of independent identically distributed random variables $X_n$ having a geometric distribution, i.e. $P (X_n = k) = p (1 − p) ^ {(k − 1)}$. Find the distribution of random variables ...
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1answer
27 views

How to prove this is a probability density function?

Being θ an aleatory variable with a uniform distribution function on the interval (-pi/2 , pi/2). Knowing that x = cos θ. How can I prove that this is a probability density function? Can anyone here ...
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18 views

Interactive Gambler's Ruin with Dash and Plotly

I created this interactive dashboard to show the concept of gamblers ruin and how you can go bust very quickly even with favourable odds. I would love some feedback: Are there any other graphs or ...
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1answer
27 views

Expected value of a function in a probabilistic game

Here is an interesting problem that I recently encountered: Suppose we have a function $f : \{0, 1\}^{n} \to \mathbb{N}$. The $2^{n}$ of the functional values of $f$ are fixed and known. Alice and Bob ...
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1answer
40 views

Three Prisioners Problem Variation

Two out of three prisoners are chosen at random to be released. Prisoner $A$ asks the guard to investigate the selected names and to tell him one of them that is not the same. Suppose that the guard ...
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1answer
35 views

question about a probability problem

i am having trouble figuring out the answer to a tricky question. i hope someone here can tell me if i'm at least on the right track here. probabilities that a part works for a year. the device has ...
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2answers
28 views

Combination Feature of Probability Question

We are given the following scenario: A nursery claims that 80% of its trees survive at least three years. As part of a service learning requirement, you are landscaping a neighborhood park and have ...
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1answer
17 views

probability theory for prob. measurement [closed]

Let $\Omega =[0,\infty)$ and prove that $\mathit P ((a,b))= \int_{a+1}^{b+1} x^{-2} dx$ is a probability measure of $\Omega$.
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Finding joint probability from coupling of Markov chains

Let $W_t,Z_t$ be Ehrenfest urn Markov chains i.e. the transition probabilities for $W_t,Z_t$ are given by $$ \begin{align} P(j,j+1)&=\frac{n-j}{n} \tag{1}\\ P(j,j-1)&=\frac{j}{n} \tag{2}\\ \...
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1answer
26 views

In an infinite coin toss, after some $k$ flips, will $k+1$ to $2k$ be heads?

Given a fair coin and infinitely many tosses, must there exist some finite integer $k$ such that after you've done exactly $k$ total tosses, you'll get heads on tosses $k+1$ through $2k$? This seems ...
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1answer
27 views

Little oh notation in probability: $P(X_n = 0)=1-\frac{1}{n}$

Let $\{X_n\}_{n \ge 1}$ be a sequence of random variables with probability distribution given by: $$P(X_n = 1)= \frac{1}{n} \quad \text{and } \quad P(X_n = 0)=1-\frac{1}{n} $$ Prove that $X_{n}= o_p(1)...
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12 views

Arrival Time and Probability with Poisson Process Word Problem

Eastbound and northbound trains arrive according to independent Poisson processes. On average, there is $1$ eastbound train every $12$ minutes and $1$ northbound train every $8$ minutes. First, I ...
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1answer
30 views

Erdős–Rényi model related problem

Several weeks ago random value $X$ Variance - Var($X$) and Expectation - $\mathbb{E}X$ were introduced in terms of our probability course. A week ago we were given problems to think about, one of them ...

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