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Questions tagged [probability]

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 (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

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1answer
26 views

False positives probability and Bayes Theorem formulation

Problem Assume a predictive model can tell from a user's Twitter posts whether the user is a male or a female. 20% of the users are females. If a person is a female, the model is 90% inclined to ...
2
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1answer
26 views

$\limsup_{n \to \infty}\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\}\subseteq \{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\leq\epsilon\}$

I recently saw an assertion made that $$\bigcap_{m \in \mathbb N} \bigcup_{n \geq m}\left\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\right\} \subseteq\left\{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\...
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0answers
21 views

non-trivial weak solution of SDE

Let consider the SDE $dX_t=|X_t|^adW_t$ for $0<a<\frac{1}{2}$  I know that this SDE has non-zero weak solution by Engelbrt & Schmidt's theorem. But I cannot find weak solution $(X,W)$ with $...
5
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1answer
35 views

Approximate Borel set by compact set

I have proven the following: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{A}$ an algebra such that $\sigma(\mathcal{A})=\mathcal{F}$. Then for every $\varepsilon>0$ ...
2
votes
1answer
56 views

Some help on Bayes Statistics/probability problem

The problem is: 95 people participated in an experiment. $N$ people were given food with an enzyme in it to help people lose weight. However, some people have gained an illness from eating the food ...
2
votes
2answers
52 views

Toy problem: How much should I rationally be willing to pay for this hypothetical and simplified insurance?

Note: This is a sub-question to help answer a larger question I've posted in the personal finance stack exchange: Rational risk-assessment decision framework: Should I buy health insurance? Consider ...
-1
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1answer
27 views

Do we have that $\mathcal{F}_{\infty} = \sigma(X_{t} \colon t \geq 0)$? [on hold]

If $(\mathcal{F}_{t})_{t \geq 0}$ is the filtration generated by a process $(X_{t})_{t \geq 0}$, one typically sets $$ \mathcal{F}_{\infty} : = \sigma \Big( \bigcup_{t \geq 0} \mathcal{F}_{t} \Big). $$...
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1answer
32 views

How do I find P(X > x + y | X > x), y > 0 when X is an exponential RV

I'm struggling to see how I would manipulate the PDF so that I could find the conditional probability? X is an exponential RV with $\lambda>0$ and the pdf=$\lambda e^{-\lambda x}$
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0answers
35 views

Question on $\liminf A_{n}$ and $\limsup A_{n}$

Say I have the following event $\{\limsup_{n \to \infty} |X_{n}|> \epsilon\}$, and that $P(\{\limsup_{n \to \infty} |X_{n}|> \epsilon\})$, I also realize that $\{\limsup_{n \to \infty} |X_{n}|&...
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2answers
29 views

How to calculate divisibility with 3 of a ratio?

Given the equation $k = (p - 1)/2$ where $p$ is any prime number, what is the chance that a randomly chosen element from the set of all $k$s will be divisible by 3? Or rather, how can this probability ...
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0answers
58 views

When is the covariance matrix $E\left(f(X)X^{\top}\right)$ invertible?

Let $X$ be a zero-mean absolutely continuous random vector in $\mathbb{R}^N$. Question: which class of continuous functions $f\,:\,\mathbb{R}^N\rightarrow \mathbb{R}^N$ yields an invertible matrix $E\...
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0answers
16 views

Probability Estimate of Winning Award

I couldn't solve the below question, please help! I need to find probability estimate that each bank winning the award. For a bank to win an award, it is based on four criteria: Capital, Profitability,...
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3answers
45 views

What is the probability of drawing two identical poker hands in a row from 1 52-card deck?

Same ranks, not same suit. Without replacement, 5 card hands, standard 52 card deck (no jokers). Example: draw A23QK draw A23QK (different suits)
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0answers
47 views

When will $f(i):=\binom{2k-1}{i}\Big((1-p)^i(1+p)^{2k-1-i}-(1+p)^i(1-p)^{2k-1-i} \Big)$ attain maximum?

When will $$f(i):=\binom{2k-1}{i}\Big((1-p)^i(1+p)^{2k-1-i}-(1+p)^i(1-p)^{2k-1-i} \Big)$$ attain maximum among $i=0,1,\dots,k-1$, for very large positive integer $k$, and $p\in (0,1)$ with $p=\Omega(...
1
vote
1answer
26 views

The steps behind finding the characteristic function of RV's under transformation

I have recently been introduced to the method to find the characteristic function of a random variable that stems from transformations of other random variables. Say, for example, $X, Y$~$\mathcal{N}(...
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0answers
26 views

PDF of a sum of independent random variables with positive-definite monotonically increasing distribution of their plus or minus values?

Below is to check my understanding of PDF's random variables: Given indepedent random variables $$ x_i=(-1,+1) \ , $$ where $i=0\cdots N-1$, and $(-1,+1)$ have equal probability of occurring, and ...
2
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1answer
25 views

Calculate $Ea^{\tau}$ where $\tau=\inf\space[{n: X_1+…+X_n=1}]$

Calculate $Ea^{\tau}$ for any $a\in (0,1)$ where $\tau=\inf\space[{n: X_1+...+X_n=1}]$ $X_i$ are independent and have distrbution as such: $P(X_i=-1)=P(X_i=1)=\frac{1}{2}$ I will add that the first ...
0
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0answers
23 views

Probability of guessing outcomes

So we want to guess the outcomes of football games. For every game there are three possibilities: - Home wins (H) - Guests win (G) - The game is a tie (T) We assume that the probability of guessing ...
6
votes
1answer
52 views

Birthday Problem without using complement

I have a solution to the birthday problem without using complements that is arriving at the wrong answer. I'd like to understand what I am doing wrong. I am not looking for alternate solutions to the ...
1
vote
1answer
19 views

probability that at least two part will be defective

Question The probability that a part manufactured by a company will be defective is $0.05$. If $15$ such parts are selected randomly and inspected, then the probability that at least two part will ...
1
vote
1answer
17 views

Probability of any family of being from a specific region.

In a building there is 95 apartments means 95 family. Suppose there is a region called $X$. For each family the probability of being of that region $X$ is $P(X) = 1/64$. what is the probability of ...
1
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1answer
38 views

Weird lottery proabablity question

In this lottery $7$ balls are chosen from $1-60$. In order to win the main prize, you must select all $7$ right. I calculate the odds of doing this as: $$1:386,206,920$$ The odds of getting $3$ ...
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0answers
28 views

If $X\sim N(\mu,\sigma^2)$, how to compute $P(\bar{X}-\mu>n^{-1/3})$?

Let $X$ be a random variable with distribution $N(\mu,\sigma^2)$. Draw a random sample of size $n$. A homework problem asks me to compute the following probability "for general $n$": $$P(\bar{X}-\...
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1answer
37 views

Joint Probability Density Functions example [on hold]

Consider the jpdf $$f(x,y)= \frac{3}{13}(xy-x^2)$$ For $x\in[0, 2]$ and $y\in[3, 4]$. Calculate the probability $y$ is less than $x+2$
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votes
1answer
32 views

Probability for cumulative density function [on hold]

I have no idea how to do this since I don't have the specific function, and prof says that it's the point in p.d.f. as it is $f(x)$. This is the problem: The random variable $x$ is distributed ...
0
votes
2answers
35 views

Finding probability when two dice are rolled [on hold]

You have two dice. Die one is a standard die with the six faces marked from 1 to 6. The second die has two faces marked with 1, two faces marked with 2 and two faces marked with 3. Both dice are ...
3
votes
1answer
49 views
+50

Show that $a_{n}M_{n} - a_{n}^{2}\xrightarrow{d} Q$

Let $(X_{i})_{i}$ be IID standard normal random variables. Define $M_{n}:=\max{(X_{1},...,X_{n})}$ and $a_{n}=\sqrt{2\log(n)-\log(\log{(n)})-\log{(4\pi)}}$ Show that $a_{n}M_{n} - a_{n}^{2}\...
1
vote
1answer
34 views

Multiple winning coin tosses in a Row, some losses, but what is total Percentage?

I would appreciate your help in the following scenario. I flip a coin and the following events happened: 3 Wins, 1 loss 3 Wins, 2 losses 5 Wins, 2 losses 1 Win, 1 loss 4 Wins, 2 losses 1 Win, 1 ...
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0answers
32 views

Probability of intersection for stars and bars

There is a vector $\vec{x}$ of length $N$, consisting of ones and zeroes. Initially $\sum_i x_i = N^x_I$. Now, each value of $\vec{x}$ that is equal to zero flips to $1$ with probability $p$. Let's ...
2
votes
1answer
21 views

Correlation between probability when order matters and not for different sample spaces

I will ask by by example Let's say we have some set of numbers from $1$ to $9$ with duplications of length $10$, e.g. $S=\{2,1,1,2,4,3,4,8,1,2\}$. Now we draw out in uniform random way $4$ numbers ...
3
votes
2answers
57 views

Expectation of observation of first value of a random variable

Given $X_k$, k $\geq 0$ be iid random variables with pmf $$p_j = P(X_1 = j), 1 \leq j \leq m; \sum_{j = 1}^{m}p_j = 1$$ Let T = min{n > $0$ : $X_n$ = $X_o$} denote the first time we observe the ...
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votes
3answers
54 views

If a die is tossed 10 times and six of the tosses are 1s, what is the expected number of 2s in the 10 tosses?

If a dice is tossed $10$ times and six of the tosses are $1$s, What is the expected number of $2$s in the $10$ tosses? I don't know if I'm overthinking this or not. Do I need to include the $6$ ...
1
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0answers
20 views

Probability of a being at a certain state after some time

A system can be in two states, $a$ and $b$. The waiting time for $a$ to become $b$ is modelled by means of an exponential distribution with parameter $\lambda,$ $1-e^{-\lambda t},$ and the waiting ...
0
votes
1answer
20 views

Indicator expected value

Let $X$~$Uc(0,1)$ You flip a coin twice with probability $X$ to get "Heads". Let $Y$ is the number of Heads you get . Let $I$ be an Indicator that get the value 1 if $Y=2$ and $0$ otherwise. What is ...
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2answers
23 views

Chose $a_n$ so that $\frac{M_n}{a_n}$ converges to 0 in distribution

Let $X_1, ... , X_n$ be i.i.d. random variables with distribution Function F and $M_n = max\{X_1, ... , X_n\}$ . Now i need to find $ a_n \in (0, \infty )$ so that $\frac{M_n}{a_n}$ convergence to 0 ...
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votes
1answer
45 views

Variance of sum of weighted gaussian random variable

This problem comes from section 3.2. at page 7 of this paper Suppose there are $N$ independent gaussian random variables $z_1,z_2,z_3,...,z_N$. That is $$z_i \sim N(\mu _i,\sigma _i^2).$$ Now let $$z=\...
0
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1answer
25 views

What is the probability that all three check into the same hotel? [on hold]

There are four hotels in a town.Three men check into hotels in the town. They can choose in $4^3$ ways. There's 4 Possibilities to choose check into the same hotel. Is it Correct $\frac{4}{4^3}$ ?
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1answer
29 views

Probability of picking the only two same cards out of 25 cards if I can only pick twice

In a set of 25 cards where two cards are only the same, what is the probability of picking those two same cards if I can only pick twice? What I only know is that at first pick it is ...
0
votes
1answer
21 views

Get 1pt if tails and 2pt if heads, what's the prob of reaching 100pts?

Consider an infinite coin toss game. You toss a single coin each toss. You get 1pt if tails and 2pts if heads. What's the probability of ever reaching 100pts in the process? Attempt: partition the ...
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0answers
19 views

Method of Transformation in Statistics

I am trying to get a handle on finding pivotal quantities to use in confidence intervals. I came across a question regarding a uniform distribution: Suppose that we take a sample of size $n = 1$ ...
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0answers
34 views

What is the likelihood that n infinite lines intersect at one point? [on hold]

1)Assuming all lines are coplanar. or 2)Assuming all lines are coplanar and nonparallel. I would like to see an answer for both if possible. If not, the more constrained outcome for #2 will work. ...
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0answers
29 views

Modeling distributed densities or is it?

Lets say that I am counting pixels over time. Each pixel is either the color red or blue. Let's say that I make up some threshold for the number of red pixels that I count and that when I reach this ...
0
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0answers
20 views

Variation combination probability

I'm trying to calculate combination or different amount of variation we can have. E.g subway have 37 million different variations of sandwiches. I'm trying to calculate similar maths for my burger ...
1
vote
1answer
54 views

A lottery will be held. From 1000 numbers, one will be randomly chosen as the winner.

A lottery will be held. From 1000 numbers, one will be randomly chosen as the winner. A lottery ticket is a random number between 1 and 1000 with replacement. How many tickets do you need to buy ...
0
votes
2answers
36 views

Finding joint pdf of $(U,V)$, where $U$ and $V$ are transformations of independent $N(0,1)$ random variables.

Suppose $X$ and $Y$ are independent standard normal random variables. Let $U = X^2 + Y^2$ and $V = \frac{X}{\sqrt{X^2 + Y^2}}$. (a) Find the joint pdf of $U$ and $V$. (b) Show that $U$ and $V$ are ...
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0answers
71 views

How many boxes of cereal do you expect to have to buy to get all N toys?

Suppose 𝑵 different toys are offered in boxes of a favorite brand of cereal. You want to collect all the different toys. How many boxes of cereal do you expect to have to buy to get all 𝑵 toys? Let ...
1
vote
1answer
25 views

How to find the probability range/scale for a histogram

I have a histogram showing the distribution of reaction times from $100$ trials worth of data. The range in times is measured in ms and ranges from $70$ ms to $420$ ms. The frequency is displayed on ...
1
vote
1answer
48 views

Rolling 9 dice probability

Nine fair dice are rolled simultaneously. What is the probability of getting three pairs? My attempt: $$P(A) = \frac{\binom{6}{3}\binom{9}{2}\binom{7}{2}\binom{5}{2}\times3\times2\times1}{6^{9}}$$ ...
0
votes
2answers
35 views

Determining $P(ABC)$ for a known $P(A), P(B), P(C)$

I'm stumped on determining P(ABC) of Part A. My understanding is: Calculate the total number of patients (100) Calculate individual $P(A), P(B),$ and $P(C)$ $(0.4; 0.35; 0.24 $ respectively) Multiply ...
1
vote
1answer
23 views

confusion on characteristic wrt convolution and product measure

I have recently come across characteristic functions. Let $X,Y$ be random variables on $(\Omega, \mathcal{F}, P)$ Let $\widehat{P_{X}}$ and $\widehat{P_{Y}}$ denote the respective characteristic ...