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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1answer
47 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 ...
0
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0answers
3 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 ...
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0answers
17 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 ...
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0answers
16 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 ...
0
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1answer
13 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 ...
-1
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0answers
18 views

Scrabble/words with friends

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 :)
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2answers
22 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
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0answers
15 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
48 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
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0answers
10 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
0answers
13 views

An optimization problem for non-homogenous poisson process with unknow distribution

Jobs arrive at an M/M/1 type server according to an non-homogenous Poisson process with rate parameter $\lambda_k$. Where $\lambda_k$ and $\mu_k$ denotes the arrival rate and service rate at $k_{th}$ ...
0
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3answers
28 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
50 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
51 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
12 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
19 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 ...
0
votes
1answer
22 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
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2answers
33 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
32 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
200 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
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5answers
109 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 ...
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3answers
32 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
268 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 ...
2
votes
1answer
43 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 ...
0
votes
1answer
327 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 ...
3
votes
1answer
46 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 ...
3
votes
1answer
37 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 $n$, ...
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 ...
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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
24 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
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0answers
27 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
31 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 ...
1
vote
1answer
31 views

Definition of standard deviation and $l_2$

If we denote the mean as $\mu$, then the standard deviation is: $$\sigma\equiv\left(\sum_{x\in X}{p(x)(x-\mu)^2}\right)^\frac{1}{2}$$ In other words, $\sigma$ is the average $l_2$ distance from $\mu$. ...
1
vote
2answers
30 views

For what fixed interest rates is a certain single-period, finite-state market arbitrage free?

A single period market with three states of nature $\omega_1$, $\omega_2$ and $\omega_3$ is given, in which a single asset is available, namely a stock that is worth $8$ units today, and whose payoff ...
0
votes
1answer
319 views

Expectation of Truncated distribution with two random variables in conditional

How to find the conditional expectation $$\mathbb E[A_1\mid A_1\ge A_m,A_2 \ge A_m,A_1+A_2 \ge 2A_y]$$ where \begin{align} 0 &\le A_1 \le 1,\\ 0 &\le A_2 \le 1,\\ \frac{1}{2} &< ...
1
vote
1answer
52 views

We have an urn with $5$ blue balls and $15$ red balls.

We remove $7$ without replacement. Let $R$ be the number of red balls removed and $B$ the number of blue balls removed. Do you expect $R$ and $B$ to be positively correlated, negatively correlated, or ...
1
vote
0answers
36 views

Probability and Uniform distribution lottery question

Suppose that a person has a lottery ticket from which she will win $X$ dollars, where $X \sim\mathrm{ Unif} (0,4)$. Suppose her utility function is $U(x) = x\alpha$ for $x \geq 0$ and $0$ otherwise, ...
1
vote
1answer
47 views

Using Feynman-Kac, compute the following:

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]$. I'm just having a hard time with ...
1
vote
1answer
32 views

“Time until arrival/departure” in a Poisson process…

Customers are served at a bank with the following process. While there is at most one customer in the bank, there will be only one person teller at a window. If a second customer comes into the ...
1
vote
1answer
22 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
-1
votes
1answer
39 views

A related problem regarding Normal Distribution (Continuous Probability) [on hold]

A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 140 feet and ...
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votes
2answers
49 views

Question on Probability 11 [on hold]

The probability that $A,B$ and $C$ can solve a problem are ${4}\over{5}$,${2}\over{3}$ and ${3}\over{7}$ respectively . The probability of problem being solved by $A$ and $B$ is $8\over15$,$B$ and $C$ ...
3
votes
4answers
53 views

Binomial distribution, given the number of success, what is the expected total number of trials?

For a random variable that follows binomial distribution, $X|N=n\sim Binomial(n,p)$. What is the expectation of $N$ when we know the value of the random variable but don't know the total? ie. What is ...
0
votes
1answer
28 views

How to find the probability of declaring faulty

My question: Consider a company that assembles computers. The probability of a faulty assembly of any computer is $x$. The company therefore subjects each computer to a testing process. This testing ...