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
26 views

Distribution of reversed k-th order statistics

Let $X_1,...X_n$ be i.i.d. Let $Y_{(i)}$ the $i$-th order statistic of that sample. The distribution function of the order statistic is given by $$F_{Y_{(i)}}(y) = \sum_{k=i}^n \binom{n}{k} y^k ...
5
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2answers
135 views

Misunderstanding the Theorem of Bayes

Saw this problem in an article on bbc.com (6 July 2014, "Do doctors understand test results?," by William Kremer BBC World Service; link below, slightly modified): A 50-year-old woman, with no prior ...
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1answer
24 views

Proper formula for this probability

I have here a probability problem that I was able to solve without using any proper formula, i just made it up myself. I wanted to know the proper formula approach for this problem: Amanda has ...
1
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2answers
43 views

Not sure with my probability understanding

I have here a problem that I am trying to solve but I am stuck somewhere and I am not sure if i am doing it right or not. Erin has some coins in her pockets. In her left pocket she has 1 nickel ...
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0answers
18 views

Confidence size and coverage probability in a confidence set?

Let $\theta \in \Theta \subseteq \mathbb{R^d}$ be the parameter of interest and let $\theta_0$ be the true population parameter value. Let $n$ be the sample size. Let $CS_n$ be the confidence set ...
0
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1answer
33 views

Probability of Random Variable Minus Random Variable

$X_1 , X_4$ ~ $ Binomial(18000,1/6)$. So $X_1+X_4$ ~ $Binomial(18000,1/3)$. I am asked to find $P(X_1-X_4)\leq 80)=?$. The solution is to find $Var(X_1-X_4)=6000$, $E[X_1-X_4]=0$ and then do the ...
0
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1answer
45 views

Need explanation about probability

I wanted to learn about probability, I have here a sample question that i want to base as my starting point. This was given as our homework but was never discussed in class on how to solve it. ...
0
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0answers
24 views

A question on probability. Is the question answerable or there is a meta logic issue? [duplicate]

I have the following question and I'm not sure of the right answer (if any), could you help me in elucidating it? If you chose an answer to this question at random, what is the chance you will be ...
0
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1answer
32 views

Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
1
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1answer
59 views

Is there a fast, reasonably accurate estimator for multinomial PDF?

I am working on a balls in boxes kind of problem, where the probability of a ball ending up in a certain box varies by box, that is, each box has some probability P of getting any ball, all together ...
0
votes
1answer
48 views

Geometric Distribution - How to show that a certain event is unplausible?

We have given a geometric distribution with parameter $p$ as well as some result $r$, which we doubt is an outcome of the given distribution. What is the best way to show that $r$ is indeed not a ...
2
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0answers
58 views

Randomness in a sequence

For a function $f$ consider a random sequence $a_{n+1}$ can be either $a_n+f(a_n)$ or $a_n-f(a_n)$ Given that the next term in the sequence is subtracting $f(a_n)$ from the previous term 50% of the ...
2
votes
1answer
40 views

Expected Payment under limited policy

The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: $$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$ ...
1
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1answer
56 views

The pdf of $X+Y$

$X,Y$ are independent. $X\sim U(0,1)$ and $$f_Y(y)=\cases{2y,\;0<y<1\\ 0,\;Else.}$$ What is the pdf of $X+Y$? (i.e. $f_{X+Y}$) I know that $$f_X(x)=\cases{1,\;0<x<1\\ 0,\;Else.}$$ But ...
1
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1answer
62 views

Delta method on sequences

We have $ \left\{X_n \right\}, \left\{Y_n \right\}$ sequences of random variables. Also $a_n \left(X_n-Y_n \right)\xrightarrow[d] {} Z$, and $X_n,Y_n\xrightarrow[d]{}\theta$. Let $g$ be continuously ...
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0answers
22 views

Combination of X1.y1 + x2, y2

Thanks in advance. Request to provide your support to solve the problem. I Have a set of X and Y combinations as follows: ...
4
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1answer
59 views

Expected maximum of a sequence of i.i.d. Poissons

Let $X_i \sim \mathrm{Pois}(1)$ be a sequence of $n$ i.i.d. random variables (with Poisson distribution with parameter 1). I'm interested in the asymptotic behavior of $$\mathbb E[\max_{i \in ...
0
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3answers
43 views

Question about normal approximation and variance

This isn't so much a question about getting a right answer as much as it's about understanding a mathematical concept, but I will give you the problem that spawned it: An analysis of data shows that ...
0
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1answer
35 views

Question about exp. distribution

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
2
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3answers
173 views

We throwing $m$ balls to $n$ cells…

We throwing $m$ balls to $n$ cells randomly... At each cell can be more then one ball, or (of course) it can still empty. What is the expectation of the empty cells? I'd like to get any help! Thank ...
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1answer
45 views

Does this recursive problem have a solution? [closed]

The variable $a$ starts at any value greater than $0$. Repeat this infinitely: $$a=a+f(a)$$ $$a=a-f(a)$$ Is there any function where $a$ will be greater than it was at its starting point?
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0answers
10 views

multivariate convergence in distribution to chi-square

Does Xn converges in distribution to Nk (0 , Ik) ⇒ transpose(Xn).Xn converges in distribution to χ2(k)? Any help will greatly be appreciated.
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0answers
28 views

Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
1
vote
1answer
40 views

probability of getting 5 calls in 5 minutes

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. 1 a. Compute the probability of receiving three calls in a 5-minute interval of time. b. Compute the ...
1
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1answer
18 views

weak convergence of probability measures and unbounded functions with bounded expectation

Assume that $\mu^n$ are probability measures on $R$ that convergence weakly(-*) to $\mu$, i.e for all $f \in C_b (R)$ (bounded and continuous), we have that $\int f(x) \mu^n(dx) \rightarrow \int f(x) ...
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0answers
14 views

Exponential Inequalities for Martingales

So I was having a read of the paper here: Exponential Inequalities for Self-Normalized Martingales. I am particularly interested in Remark 4.2, which states that if $M_n$ is a Gaussian martingale (and ...
0
votes
1answer
26 views

Need help with P(D) in a Bayesian model

So I've been reading about Bayesian models so I tried I'd have a toy example I could play with. Consider the following: You are at a bus stop and you observe the bus arriving at various times $t_1, ...
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2answers
29 views

What is Cumulative Binomial probabilities?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ...
0
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2answers
284 views

Two groups A and B are playing a game…

Two groups A and B are playing a game. The first group that wins 3 times is the winner. The probability that group A will win at on game is $\frac12$ and the same thing for group B. $X$ = The number ...
6
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0answers
180 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
2
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1answer
35 views

CDF for non-homogeneous Poisson process [duplicate]

I am trying to understand the inverse transform method for simulating random processes and have managed to completely confuse myself. Consider a Poisson process whose conditional intensity is ...
1
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3answers
79 views

Why Maximize Expected Value?

In many instances I've come across (in Game Theory, etc), when trying to choose an optimal strategy it has the criterion that it wants to maximize expected value much of the time. To simplify this ...
0
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0answers
42 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
2
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1answer
42 views

What is the “average time remaining” when guessing a random value?

I lack the terminology to ask this question "properly", so to illustrate what I'm bouncing around in my mind, let's take a story example: John wrote a script which guesses passwords. He has a list ...
3
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0answers
63 views

Coin flip: Double or Nothing

I have an amount of chips to bet on a fair coin flipping and landing on heads. For each time in a row that it does land on heads, the amount of chips in the bet is doubled by the house, and I am able ...
6
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3answers
567 views

Birthday Problem - Company Stats Strange or Average?

I had a birthday problem question that I'm really interested in knowing the answer for: In a group of 2,000 people, what is the probability of one day during the year that no one has that particular ...
2
votes
1answer
93 views

Tough combinatorics problem

We have an urn containing $n_a$ tiles labelled "A", $n_b$ ones labelled "B", and $n_c$ tiles labelled "C". We also have a string of letters consisting of $s_a$ occurrences of the letter "A", $s_b$ ...
1
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1answer
75 views

Convergence in Probability of maximum of r.v.

Suppose that $\{X_n^{(1)}\}, \ldots, \{X_n^{(k)}\}$ are sequences of random variables that $X_n^{(i)}\rightarrow_p 0$ as $n\rightarrow \infty$ for each $i=1,...,k$. I have to show that $ \max_{1\leq ...
0
votes
2answers
76 views

please help me ( probabilities )

please let me know if my answer true or false Three numbers are chosen at random without replacement from the set {0, 1, 2, 3, ... , 10}. Calculate the probabilities that for the three numbers drawn ...
1
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2answers
53 views

Efron-Stein inequality

The Efron-Stein inequality sais that if $X_1,\ldots,X_n$ are independent random variables on say $R^n$, and $f:R^n \rightarrow R$ s.t. $Z:=f(X_1,\ldots,X_n)$ has finite variance, then ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
0
votes
1answer
32 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
1
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2answers
35 views

Counting exercises - Solution verification.

i'm studying some combinatorics and i came up in the following exercises. Suppose we are given a set $U$ of $n$ elements. Suppose $A \subset U$ has $k$ elements. Determine the number of subsets ...
0
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1answer
55 views

Measure extension theorem(unique) [closed]

Please give an example of two probability measures $\mu \not = \nu$ on $\cal{F} $= all subsets of {1, 2, 3, 4} that agree on a collection of sets C with $\sigma(C)=\cal{F}$ . thanks in advance.
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2answers
32 views

On the definition of a random variables

Let $(O,F,P)$ be a probability space. That is $O$ is a set, $F$ is a $\sigma$-algebra of subsets of $O$ and $P$ is a probability measure. Consider a function $f:O\to\mathbb R$. Would we call $f$ a ...
1
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1answer
39 views

Find Limiting Distribution of $|X_n|$

Let $Z_1,Z_2,...,Z_n,...$ be a sequence of independent standard normal random variables. Let $X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}$. Does the limiting distribution of $|X_n|$ exists? If yes, find it; ...
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2answers
36 views

Computing Probability

Suppose there are 10 students , out of which 6 are selected. Assume I have a list of 5 students with me. What is the probability that AT LEAST 3 students from my list are among the 6 students ...
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3answers
60 views

What is the Probability? [closed]

You randomly remove three fish from a tank. How many fish in the tank will ensure greater than 50% chance of having a male and a female assuming the tank originally has an equal number of males and ...
0
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1answer
60 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
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2answers
31 views

We are making a Bernoulli experiment… [on hold]

We are making series of independent Bernoulli experiment with $\frac13$ chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ...