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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3 views

Introductory study of Survival Analysis and Decision Theory

I'm pursuing a compact Masters degree in Mathematics, a 4 year program at BITS Pilani, India. Except for a couple of introductory courses on statistical and probabilistic analysis, and operations ...
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14 views

Deducing means of normal distributions

I have a set of objects ($i$) with different lengths ($x_i$), and two devices ($a$ and $b$) to measure them. Each device measures with an error which probability density function (pdf) follows a ...
3
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2answers
18 views

Why is $ -\sum_{i \in \text I} p_i \log_2(p_i)$ maximized for all $p_i$ equal? Is it true if $|\text I | = \infty$?

Reading a text it is stated without proof that $$ -\sum_{i \in \text I} p_i \log_2(p_i)$$ where $\sum_{i \in \text I} p_i = 1$ is maximized if $p_i$ is a constant. In the case of my theorem, the ...
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0answers
10 views

Elk recapturing experiment: Expected size of sample till $m$ tagged elk captured

Out of a population of $N$ elk in a forest, a random sample of $n$ elk are captured, tagged and released. Later, a new sample of size $m$ is randomly drawn. a) Find the PMFs of the number of ...
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2answers
37 views

Examples of combinatorial/probabilistic proofs of theorems in linear algebra

Are there any examples of combinatorial/probabilistic proofs of theorems in linear algebra? Motivation: I see here, the inverse is true.
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1answer
23 views

Question regarding solution to the problem of computing the expected number of coin flips until getting 5 in a row.

I have a question for further explanation regarding the solution to the problem here Expected Number of Coin Tosses to Get Five Consecutive Heads. The most favorite solution by André Nicolas is: ...
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1answer
12 views

Distribution of a Gaussian Random variable vector [on hold]

I read a slide on the internet which show that: If the random vector $w\sim N(0,I )$ then how can I prove: $x= A^{1/2}w+\bar{x}$ has the distribution $N(\bar{x},A)$ Here A is the covariance matrix ...
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0answers
15 views

Suppose X follows a binomial distribution with parameters n=100 and p=1/5,then prove that P(X=r) is maximum when r=33 [on hold]

smeone plz solve this question Q.Suppose X follows a binomial distribution with parameters n=100 and p=1/5,then prove that P(X=r) is maximum when r=33. aren't we required to find the mode here which ...
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0answers
26 views

Textbook RSA game with one prime

Let p be a n-bits prime number, that is drawn uniformly. Let e and m uniformly drawn from Z(p-1) and Z*(p) respectively. Let y= (m^e) mod p Prove that the probability to find m while knowing only ...
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0answers
26 views

Negative binomial distribution [on hold]

Ann plays a casino game and receives a token whenever a six or a one comes up when a die is rolled. The games ends when she get y tokens ; she receives x dollars where x is the number of rolls made. ...
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35 views

Need formula to calculate probabilities for a forex trading system [on hold]

I'm developing a forex trading system and I'm trying to understand how to calculate the probabilities involved. Unfortunately, my math skills are lacking. Here are the system basics: The stop loss ...
3
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0answers
50 views

Probability of Posting a Quad and Trip on 4chan

Important Pre-Requisite Knowledge On the image board 4chan, every time you post your post gets a 9 digit post ID. An example of this post ID would be $586794945$. A Quad is a post ID which ends with ...
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0answers
12 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them?
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0answers
39 views

Strong law of large numbers for square-integrable and uncorrelated random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable and uncorrelated (maybe we actually need independence) random variables $\Omega\to ...
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1answer
29 views

Football pool question

I'm in a football pool where before the game is played, we pick numbers from 0 to 9 from a bag. The winner is whoever picks the number that is the sum of the last number of the two scores. Eg. 32-27, ...
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2answers
39 views

Let $X$ be Hypergeometric, Find $E\left(\binom{X}{2}\right)$

Let X be Hypergeometric: $X \sim \operatorname{HGeom}(w,b,n)$, so that $X$ is the number of white balls in a sample of size $n$ out of a population of $w+b$ white and black balls. Find ...
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2answers
67 views

Using probability to detect exam cheating (identical wrong answers)

Hypothetical: What’s the probability that two people taking a test with 10 questions get the identical wrong answers? (Let's say there are 4 choices per problem) Should we first break this down ...
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0answers
21 views

Card Matching: expected value of correctly predicted cards with partial feedback

A standard deck of cards is shuffled, and the cards are dealt face down one by one. Just after each card is dealt, you name any card (as your prediction). Let X be the number of cards you predict ...
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4answers
88 views

Monty Hall problem (shifting probabilities)

I was explaining the Monty Hall problem to someone, and I explained it in this way: You have three doors, and you pick one, giving you a $1/3$ chance of being right. The presenter opens one of the ...
0
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1answer
28 views

Getting P-value of test; statistics

In order to test $H_0 : \mu = 50$ vs $H_{\text{a}} : \mu \neq 50$, a random sample of 9 observations (from a normally distributed population) is obtained, yielding $\bar{x} = 61$ and $s = 21$. What is ...
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1answer
27 views

Probability and range partition

in this question we have a fixed partition and we want to partition the range to obtain a three subsets with the condition below.
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0answers
21 views

Evaluate the sum $n$ of geometric random variables

Let $X_i\sim G\left (1-\frac{1-i}{n}\right)$. Evaluate $ \sum_{n=1}^n X_i$ My Try: $$ \sum_{i=1}^n X_i = \sum_{i=1}^n \sum_{k=1}^\infty \left(\frac {i-1}{n}\right)^{k-1}\left( 1 - ...
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2answers
26 views

Show that $\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$

Let $X,Y$ two random variables of the same probability space. Show that $$\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$$ I think I need to start from LHS and split it ...
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0answers
29 views

Three state probability where one state has yet to factor results. [on hold]

I'm currently trying to explain something to someone else using probability to make it simpler to understand. As I have it now there have been 5 examples that have happened. In one case there are 3 ...
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1answer
75 views

I do not understand the last step of this proof. [on hold]

1. PLEASE LOOK THE FOLLOWING PROOF FIRST. 2. Suzu explained the fist several steps to me in this page :Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$ . But I still ...
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0answers
38 views

Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
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3answers
92 views

Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?

How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??
4
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1answer
91 views

How to write $1-x-x^3+x^4+x^5+x^6-x^7 \cdots$ as a power series representation

How can I write $1-x-x^3+x^4+x^5+x^6-x^7 ....$ as a power series representation (i.e., a neat fraction such as $\frac{1}{1-x}$. This stems from $\binom{\text{number of partitions of }n}{\text{into an ...
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0answers
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An example of memoryless yet non-independent random process?

I am new to random process. I know that independence indicates memorylessness yet the memorylessness is not necessarily independence. There are abundant examples of independent random process (like ...
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1answer
26 views

If $X\ge 0$ and $a\ge E[X]$, then $P(X\gt a)\ge (E(X)-a)^2/ E(X^2)$ [on hold]

I need help with this problem. Prove that if $X\ge0$ and $E[X^2]<\infty$ then for all $a\neq0$, $E[X]\le a$, we have $$P(X\gt a)\ge\frac {(E(X)-a)^2}{E(X^2)}$$ Progress I have my doubts if ...
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0answers
23 views

Pollaczek-Khinchin formula for ruin probability - proof

I got stuck in a specific part of proof of the Pollaczek-Khinchin formula (in book "Stochastic Processes for Insurance and Finance", T. Rolski et al., section 5.3.3, theorem 5.3.4). Namely, why the ...
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1answer
32 views

Adjacent dominos in a train

Definition of a domino -- a domino contains two squares separated by a line. In both of the squares, there are some numbers of dots (can be 0). Definition of "double-n" domino set: It contains one of ...
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1answer
31 views

Identifiying Biased and Unbiased Samples

My little nephew asked me a question about biased/unbiased samples in which is teachers answer is something I disagree with to say the least (I don't agree with the assumption made by the teacher nor ...
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1answer
31 views

Probability of drawing an element from a countably infinite sequence

Consider a sequence containing $A$ and $B$ where, starting at $n=0$, there are $2^n A$'s followed by $2^{n+1} B \ $'s, so the sequence begins $$A, B, B, A, A, B, B, B, B, A, A, A, A, B, B, B, B, B, ...
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1answer
29 views

What is the probability of that event? [on hold]

A fair coin is tossed repeatedly. What is the probability of the event "Three consecutive heads occur before two consecutive tails"?
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1answer
40 views

Expected number of matching “cards”. Why is $\sum_{m=0}^n D_{n,m} = \sum_{m=0}^n m \cdot D_{n,m}$?

Each of n ≥ 2 people puts his or her name on a slip of paper (no two have the same name). The slips of paper are shuffled in a hat, and then each person draws one (uni- formly at random at each ...
5
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4answers
287 views

Find the fraction where the decimal expansion is infinite?

Find the fraction with integers for the numerator and denominator, where the decimal expansion is $0.11235.....$ The numerator and denominator must be less than $100$. Find the fraction. I ...
1
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1answer
17 views

Random Walk With Absorbing Barrier [on hold]

Consider a random walk $S_{t}$ with a lower absorbing barrier at $0$, and no upper absorbing barrier. $$ {\mathbb P}\left(\, S_{t + 1} - S_{t} = 2.5\,\right) =0.5\,,\quad\mbox{and}\quad{\mathbb ...
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1answer
24 views

$P(X=c)=0$ for normally distributed $X$?

Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?
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0answers
37 views

A treatise on Probabilistic arguments and Laplace/Fourier transforms to solve limits/integrals from basic calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those ...
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0answers
43 views

Calculate expected values of the lengths of line segments

There is a line segment of the length of $1$. $N-1$ points are randomly chosen in it, so it is divided by $N$ parts. The question is to calculate expected values of all these parts, from the shortest ...
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1answer
12 views

definition of Cumulative distribution function

let X be RV, and his Cumulative distribution function: there is a difference if in my case if $X<x$ ? the definition is the same?
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3answers
116 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard F. Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
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1answer
36 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ (The limits of the integrals are all ...
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3answers
51 views

Probability when n balls put randomly in n boxes such that each box contain 1 ball [on hold]

There are 100 boxes in front of you. You have 100 balls in your pocket which you throw one by one towards the boxes in front of you. Each ball will definitely end up in a box and has equal probability ...
2
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2answers
34 views

Probability in dice, Feller exercise

I am stuck with exercise 2 of Chapter 4 Feller vol 1 "an introduction to probability theory and its application". Here I report the exercise text: Five dice are thrown. Find the probability that at ...
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0answers
18 views

Probability (Dependent Events) [on hold]

The M.com class consists of 60 students, 12 of them are girls and 48 boys, 10 of them are rich and 50 not, 15 of them are fair complexioned. What is the probability of selecting a fair complexioned ...
2
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1answer
22 views

Derivation of the third moment of Poisson distribution using Stein-Chen identity

(a) Use LOTUS to show that for $X \sim \operatorname{Pois}(\lambda)$ and any function g, $E(Xg(X)) = λE(g(X + 1))$. This is called the Stein-Chen identity for the Poisson. (b) Find the third ...
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3answers
46 views

Where does this conditional probability law come from?

I was trying to follow a computation done in my class notes, and was having difficulty seeing the inspiration for a part of the manipulation in a question regarding probability. I did some Googling, ...
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0answers
13 views

Deviation of number of cycles of length 4 in Erdős–Rényi random graphs.

I'm working on my homework and can't find any relevant information for this problem. Problem: Let $G(n, p)$ be Erdős–Rényi random graph. I need to find deviation of number of cycles of length 4 in ...