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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0
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0answers
9 views

What is the proof behind the mean confidence interval for a Binomial Distribution?

How do we obtain the range to be as [$\mu-$$zσ$, $\mu+$$zσ$]? Is it when $n$ is sufficiently big?
4
votes
2answers
57 views

How to take into account uncertainty on number of events

Suppose I generate a set of events $X_{i}$ for $i = 1,2 \dots N$ and suppose every event is either a success or a failure, ie. $X_{i} = 0, 1$. If $N$ is fixed, the MLE for the probability of success ...
2
votes
0answers
12 views

Probability of a run of *k* or more of a subset of categories in *m* multinoulli trials?

Given a multinoulli distribution of categories $(C_1,C_2,...,C_n)$ with associated probabilities $\left\{p_1,p_2,\ldots ,p_n\right\}$ with $\sum _{i=1}^n p_i=1$, is there a tractable way to get the ...
0
votes
1answer
51 views

Coin Toss Experiment

I conducted an experiment where I tossed a coin $n=100$ times. I am assuming that the coin flips heads with a probability $p=0.5$. So that the coin is fair with a level of significance of $5%$, I want ...
3
votes
1answer
50 views

Probability question from GRE subject test

I ran across the following problem while practicing for the GRE math subject test: Suppose $X$ is a discrete random variable on the set of positive integers such that for each positive integer $n$, ...
0
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1answer
12 views

Extreme value distributions of unaccountably infinite set of random variables

Let us suppose that we have an uncountably infinite set $A=\{x_1,x_2, \cdots\}$ of i.i.d. random variables $x_i$, say with gamma distribution. Are minimum and maximum extreme value distributions ...
0
votes
1answer
62 views

What is the probability of an event happening in some interval given probability of it in x interval?

Suppose there is an event that happens with a probability of y in x interval of time, what would be the probability of it happening in x/2 interval of time? Would that be y/2 or is there something ...
2
votes
1answer
46 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
2
votes
2answers
53 views

Probability distribution of number of waiting customers in front of a counter

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
3
votes
2answers
71 views

Combinatoric Birthday Paradox

There is likely a closed form solution for this problem but it's had me puzzled for days. This is about a variant on the classic birthday paradox. To recap, the birthday paradox is where given only 23 ...
-1
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0answers
27 views

Odds for rolling specific faces on a 3-sided die [on hold]

Firstly, thank you for taking my question! Imagine five (5) $3$-sided dice, so three unique faces. What are the odds (percentage) of rolling $3$ of the same face when rolling $5$ dice? When rolling ...
1
vote
3answers
131 views

$\mathrm E [X \mid X=x] = x$?

I've gotten so caught up in measure-theoretic probability that I'm actually having trouble showing this simple result. Let $X$ be an integrable random variable. Then $$ \mathrm E[X \mid X=x] = ...
-5
votes
1answer
26 views

lottery question [on hold]

for the lottery -- if I have 4 numbers How can I see all of the 4 number combinations, never using the same number more than once in each combination
2
votes
3answers
274 views

Is the limit of càdlàg functions càdlàg?

Is the pointwise limit of càdlàg functions càdlàg? If not which are the weaker conditions to assure it? I cannot find a counterexample
-3
votes
2answers
46 views

100-sided dice was rolled 98 times, how do you choose next numbes to bet, based on current outcomes.

100-sided dice was rolled 98 times, Numbers form 1 to 50 were rolled exactly once, except number 25, which wasn't rolled yet. Number 75 was rolled 49 times You can only bet if the next roll result ...
0
votes
2answers
219 views

Probability of no more than three heads given that at least one toss resulted in heads

You toss a coin four times. Find the probability of no more than three heads given that at least one toss resulted in heads. So if I set event A as no more than three heads and B as at least one ...
1
vote
0answers
43 views

When the sum of Markov chains is a Markov chain: “dumb” algorithm

Suppose I have two (independent) discrete-time and space, preferably non-homogeneous Markov chains $\Gamma^{(i)}=\{\gamma_1^{(i)},\gamma_2^{(i)},...\}, \ i=1,2$ and I want to find a way to check when ...
0
votes
1answer
29 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
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0answers
78 views

Optimal allocation in network

We want to analyse specialization matters in a given network (N,g). Nodes represent individuals that can produce goods and services (just like in our usual economy) and that can be consumers too. ...
0
votes
1answer
20 views

Is there an upper bound for expectation of product of two measurable function on a random variable?

I wonder if there is an useful upper bound for $\mathbb{E}_{x\sim p(x)}[f(x)g(x)]$ in the following form: $$ \mathbb{E}_{x\sim p(x)}[f(x)g(x)] \leq \mathbb{E}_{x\sim p(x)}[f(x)]\times xxxxxx $$ The ...
-2
votes
0answers
46 views

Probability - There is a radar, a computer and a gyroscope

There is a radar, a computer and a gyroscope on board an airplane. The probability that the radar fails is 0.2. If the radar fails, the gyroscope will also fail, and the probability that the computer ...
-1
votes
1answer
15 views

Distribution of specific distributions

I have a normal distribution of independent variables, and there are a specific number of samples to this distribution: say 1 million samples. A function is set by the largest value of these million ...
-2
votes
0answers
21 views

Scrabble/words with friends [on hold]

How many letter combinations are possible with 7 tiles? Just the math answer please, 7 tiles in 7 slots, how many different combinations? Thank you :)
0
votes
2answers
27 views

Why is this counting function finite? (It is used Probability)

Why is this counting function finite? I don't understand this interpretation of the author. Can you explain more about this? Please.
20
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5answers
2k views

Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
0
votes
0answers
23 views

Distribution of the test statistic?

Let $\mathbf{x}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma)$. I am trying to find a distribution of the following test statistic $ T(\mathbf{x}) = \frac{\bar{\mathbf{x}}^H ...
-1
votes
1answer
50 views

Billingsley Exercise 8.8 (Markov Chains)

I am studying from Billingsley and would like some hints on the following exercise. Suppose $S = \{0,1,2,...\}$, $p_{00} = 1,$ and $f_{i0} > 0$ for all $i$. Here, $S$ represents the state ...
0
votes
0answers
12 views

Meaningful Extreme value distribution

Extreme value theory (EVT) dictates that the limit distribution of the minimum of the set of i.i.d. Chi-square random varibales $\{C_1,C_2,\cdots,C_n\}$ is Weibull. The Weibull distribution has ...
0
votes
3answers
33 views

Probability of picking a card one out of 52 times.

Let's say we have a standard deck of 52 cards. What would be the probability of choosing the 2 of diamonds? Obviously, it would be $\frac{1}{52}$. If we were to randomly choose another card from ...
2
votes
2answers
54 views

What conditional independence theorem is being used here

In stanford's machine learning lecture 1, linear regression is defined on page 11, section 3 as: For $i = 1, \ldots, m$, $y^{(i)} = \theta^T x^{(i)} + \epsilon^{(i)}$, where $\epsilon^{(i)}$ are IID ...
2
votes
0answers
53 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
1
vote
1answer
13 views

Relationship between minimizing a conditional variance and a covariance

We are working with discrete-time stochastic processes. Let $v_k$ be a $\mathcal F_k$-predictable process, and let $X_k, \eta_k$ be $\mathcal F_k$-adapted processes. Define $V_k = v_kX_k+\eta_k$ and ...
0
votes
1answer
23 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
0
votes
1answer
289 views

How to Calc the odds of winning a lucky dip

How do I calculate the odds of winning? I am doing a lucky dip raffle - you pay $£1$ and pick out $3$ balls, there are $495$ balls and $50$ prizes. Each ball has a number on, and if the number matches ...
1
vote
2answers
36 views

Find the following probability

A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue. If four chips are taken at random and without replacement, find the probability that there is at least 1 chip of each colour. ...
0
votes
1answer
19 views

How are Chi Square probabilities calculated?

What steps would one follow to calculate the values in a Chi Square probability table such as https://people.richland.edu/james/lecture/m170/tbl-chi.html? Say you had 15 degrees of freedom and wanted ...
0
votes
1answer
102 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
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votes
2answers
35 views

How to solve this probability formulation? [on hold]

$\int_{200}^{250} P(a=x \land 450-x \leq b \leq 250)\space dx$, where $a$ and $b$ are uniformly distributed random variables on $(0,250]$ and $(10, 250]$ respectively.
1
vote
1answer
201 views

Generalization of Bayes' Theorem

Does anyone know of a generalization of Bayes' theorem to multiple conditions? From this answer I can see the definition of conditional probability with multiple conditions, but I couldn't find any ...
3
votes
5answers
120 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
0
votes
3answers
33 views

Confused about definition of absorption probability

My confusion can probably most easily be explained with an example. Consider the following one step transition matrix : $$ P=\matrix{% & 0 & 1 & 2 & 3 & 4 \\ 0 & ...
1
vote
2answers
269 views

Probability space for stochastic processes

In Sinai's book on stochastic processes, the definition for discrete time stochastic processes is "a sequence of random variables $\{X_{n}\}_{n\in{}T}$ defined on a common probability space ...
0
votes
1answer
328 views

Characteristic function and probability density function: Fourier or Inverse Fourier?

I have come across two contradicting definitions of characteristics function (CHF). In wikipedia CHF is defined as the inverse Fourier transform (FT) of probability density function (PDF) and at some ...
5
votes
3answers
8k views

If three dice are rolled, what is the probability that all three are the same number?

The dice are fair. You have a $1\over6$ chance of getting the first number. A $1\over6$ chance of the second and so on. Is it just $({1\over6})^3$ (1/216) or is that not accounting for the second ...
12
votes
1answer
297 views

Shooting bullets

This is from http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/May2014.html Every second, a gun shoots a bullet in the same direction at a random constant speed between 0 and 1. The ...
-1
votes
0answers
33 views

How to calculate $P(X_1 < X_2 < X_3…X_n ) $ [on hold]

Could you please help with the following problem i am having- I need to calculate the probability of $X_1$ (randomly selected discrete value between $a$ and $b$) being smaller then $X_2$ (randomly ...
2
votes
2answers
25 views

Distribution of a product of Multinomials

Consider the following: $(X_1, X_2, X_3, X_4) \sim \mathrm{Multinomial} (n,\mathbf{p})$ where $\mathbf{p} = (p_1,p_2,p_3,p_4)$. I would like to find the distribution of $X_1 X_4$, or at least know ...
0
votes
0answers
29 views

Show that the following set function is not a probability set function

If the sample space is $\mathfrak{C} = \{c : -\infty < c < \infty\}$ and if $C \subset \mathfrak{C}$ is a set for which the integral $\int\limits_C e^{-|x|}dx$ exists, show that this set ...
4
votes
1answer
32 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Probability: Theory and Examples (4e), Theorem 8.4.1, page 317), there's a step which I'm a little shaky on. Basically, this proof ...
4
votes
2answers
59 views

Probability question related to coin tosses

In an exam I gave recently, the following question was asked: A fair coin is tossed $10$ times and the outcomes are listed. let $H_i$ be the event that the $i^{th}$ outcome is a head and $A_m$ be the ...