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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Proof of the inclusion-exclusion formula in probability

Let $(\Omega,\mathcal F, P)$ be a probability space and let $A_1,A_2,...,A_n$ be events in $\mathcal F$. Prove the following inclusion-exclusion formula $P(\bigcup_{i=1}^nA_i)=\sum_{k=1}^n$ ...
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3answers
21 views

If probability density functions converge a.e., then cumulative density functions converge

I have read a conclusion in a textbook: Suppose $f_n,f$ are density functions of some r.v. also $f_n\to f$ a.e., then $$\int f_n \mathrm{d}x \to\int f \mathrm{d}x $$ Fisrt I want to use ...
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1answer
11 views

Expanding the expected value

How to expand: E(Y+1)^2 my working out: E(Y^2)+E(1^2) = E(Y^2)+1 (I'm not sure why this is though..) Can someone link to or list the rules for expanding the expected value ......
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0answers
10 views

Sinchronicity and probability

"Synchronicity is the experience of two or more events as meaningfully related, where they are unlikely to be causally related. The subject sees it as a meaningful coincidence." Wikipedia First, I am ...
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2answers
37 views

Maximizing heads/number of flips game

Flip a coin until you wish to stop. Your goal is to maximize the ratio number of heads/total number of flips. What is the expected value of this game? Additionally, how would one play this game?
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2answers
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Finding values of a constant in a probability distribution

A probability distribution for the random variable $X$ is defined by: $$\mathbb{P}[X=x] = K\cdot(0.9)^x,\quad x = 0,1,2,\ldots$$ It is asked to find $\mathbb{P}[X\geq 2]$. When there is a domain for ...
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1answer
29 views

Expectation of uniform distribution with unknown parameter, given maximal (minimal) observation.

Let $x_i \text{ be} ~ i.i.d. ~ \sim Uni[0,\theta]$ $(\theta \text{ unknown})$. Denote $M_n = \max x_i$. So, through circumferential means, I can show that $E(x_1|M_n) = \frac{n+1}{2n} M_n$. The ...
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1answer
28 views

Find some probabilities given the probability tree

i've been practicing probability since it's not my strength, but i am doing that without a tutor or an official course, just books and videos. I was reading a problem, and i was capable of draw the ...
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0answers
8 views

Density function of $(X_{(1)}, X_{(2)})$ where $X_i$ ~ Exp(1)

I'm reading an answer to a question, which quotes a (non-English) book that I don't have access to. I'm wondering if someone here can shed some light on this claim: Let $X_1, X_2, ...$ independent ...
4
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1answer
121 views
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Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
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1answer
248 views

Probability matching problem

Arriving at party n guests throw their hats into pile. When they leave they each take a hat that is chosen randomly from the pile. We want to compute the probability of the event $A$ that at least one ...
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3answers
57 views

Plausibility vs Probability

http://whatho.in/2013/plausibility-versus-probability/ refers to pp 155-156 of 533 of Thinking, Fast and Slow by Daniel Kahneman. I'll use one of Kahneman's other questions from p 156: A ...
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0answers
12 views

Multivariable chain rule and an integral function

Can the chain rule be used in the following scenario? Let $u(r,t) = \int g(r,t)~f_R(r)~dr$ where $f_R(r)$ is a well-defined probability distribution. Then we can write $u(r,t)=u(x,y)=\int x~y~dr$ ...
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10answers
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Should I throw the dice again if I have rolled 4?

My math skills are very basic so it might be a stupid question, I had a discussion with my brother in law and now we have a 'math problem'. We were playing a game with dices and he threw 4. The ...
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1answer
28 views

How to compute this conditional probability in Bayesian Networks?

I met a problem related to conditional probability from the article "Bayesian Networks without Tears"(download) on page 3. According to the Figure 2, the author says $$P(fo=yes|lo=true, ...
4
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1answer
50 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
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3answers
50 views

Are the absolute values of random variables iid if the random variables are iid?

If $X$ and $Y$ are independent and identically distributed (iid) random variables, does it imply that $|X|$ and $|Y|$ are iid? How would you go about proving this?
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2answers
53 views

What is the expected value of A?

The Happy Animals Kennel has 18 cages in a row. They allocate these cages at random to 6 dogs, 6 cats, and 6 pot-bellied pigs (with one animal per cage). All arrangements are equally likely. Let A ...
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0answers
9 views

Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
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1answer
22 views

Proof that a random variable has exponential distribution.

Supose that $X_1$ is a continuous and positive (real) random variable with exponential distribution, namely $$P(X_1>t)=e^{-\lambda t}\quad t>0$$ Now suppose that $X_2$ is another continuous and ...
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0answers
11 views

Ross probability models questions [on hold]

I am studying for a course and have no professors to talk to live, so I hope some members here can be kind enough to help me. Rather than writing everything out, and splitting it up into different ...
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2answers
348 views

What is the expected value of the number of circles formed?

There are $99$ identical square tiles, each with a quarter-circle drawn on it. When the tiles are randomly arranged in a $9$ by $11$ rectangle, what is the expected value of the number of full circles ...
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3answers
185 views

Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$

I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$ I know that $$2^n = ...
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1answer
210 views

Variance for sum of two correlated normal random variables

$\newcommand{\Var}{\operatorname{Var}}$ $\newcommand{\Cov}{\operatorname{Cov}}$ I know that $\Var(X+Y)=\Var(X)+\Var(Y)+2\cdot \Cov(X,Y)$ but what about $\Var(X \mid x>0 + Y)$?
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2answers
64 views

Brainteaser: Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. Expected number of games before one is bankrupt?

Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. What is the expected number of games they play before one is bankrupt? I have struggled at this for hours now with little ...
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2answers
60 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
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0answers
25 views

Probability for sparse matrix after permutation

I need to calculate the probability of the following question, it is kind of tricky but I cannot work out the exact value. For a sparse (most of its entries are zero) matrix $X=[x_1, x_2, \cdots, ...
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2answers
719 views

cdf/pmf/pdf validity question

Studying for a statistics exam. I have come across this problem: and it presents to me some important and extremely basic questions (I have a LONG way to go before I'm prepared for this exam). ...
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1answer
35 views

If the two-engine plane cannot take off unless both engines are operating properly, which plane is safer on takeoff?

I am practicing a bunch of probability problems I find through random sources and I am stuck with this one. Suppose the probability that the engine in a single-engine fighter will fail on take-off is ...
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1answer
27 views

Calculate probability and its accuracy from results of experiments

You have a machine that tells you which of two items weights more than the other. You insert one object in slot 1 and the other in slot 2, press a button and then the machine tells you either "Item in ...
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2answers
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What is the expected value of the number of anchors of $S$?

For any subset $S\subseteq\{1,2,\ldots,15\}$, call a number $n$ an anchor for $S$ if $n$ and $n+ |S|$ are both elements of $S$. For example, $4$ is an anchor of the set $S=\{4,7,14\}$, since $4\in S$ ...
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3answers
81 views

Probability that the red fish are the first species to become extinct

I have a doubt in the solution of the next problem: A pond contains $3$ distinct species of fish, which we will call the Red, Blue, and Green fish. There are r Red, b Blue, and g Green fish. ...
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0answers
30 views

How is this paper using probability notation?

I am trying to understand this paper about documents and sentences. At the end of page three, they say: Let g(wi, wj ) be the distance between two events (1 if in the same sentence, 2 in neighboring, ...
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1answer
59 views

Variation of “The secretary problem”

I was assigned to write a computer program that simulates a CPU, but it is more like a game: A queue of processes is initialized: $P_1,P_2...P_n$, ordered in some random permutation: $P_{\sigma ...
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2answers
370 views

Finding the distribution of a poisson distribution with random variable lambda

So suppose $X$ is a rv with a Poisson distribution with $\lambda$ being a random variable as well. $\lambda$ has an exponential distribution with mean $1/c$ and $f_\lambda(t) = c\times\exp(-ct)1_{[0, ...
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3answers
26 views

Probability of getting a certain group of students when choosing three at random out of 25

A teacher randomly chooses a group of three students from her class of 25 students. Find: a) Probability that friends Suri, Lily and Violeta are chosen for the group? b) If he ...
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1answer
24 views

Fudge Dice: Reroll vs. Bonus

A "fudge die" is a die with equal probability to result in -1, 0, or +1. The commercially produced fudge dice are generally 6-sided dice with two "–", two "+", and two blank sides. In the ...
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0answers
21 views

Expectation of a logarithmic/trigonometric function

I am trying to find a closed form solution of the following expectation: $$\mathbb{E}[\log(a+b\cos(\phi))]$$ where $a$ and $b$ are real constants, and the expectation is with respect to $\phi$. If ...
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1answer
33 views

Mean of max vs max of mean

If I have say an $n$ collection of 10 random variables $X_1, \ldots, X_{10}$ (so an $n \times 10$ matrix of values) from some underlying distribution whether Gaussian or uniform, and I calculate ...
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1answer
33 views

A Bayesian estimate of the bias of a coin

Consider a coin with unknown probability $p$ of landing on head. I will toss the coin and stop as soon as I get a head. Say this is after $n$ tosses. If my prior belief for $p$ was uniform on ...
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1answer
22 views

CDF of random variables

due to my lack of knowledge in probability theory, I have first to apologize if the following question is not formulated in a proper language. I was wondering if there is any formal expression of the ...
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0answers
12 views

Check work for finding Max log-Likelihood of a geometric Distribution

Here is my geometric distribution: $P(L=n)=p^{n}*(1-p)$ To find the max likelihood, I do: $\sum_{L_i} L_i\log(p) + \log(1-p)$, where L_i is a particular length. I take the derivatives and end up ...
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1answer
54 views

Kelly criterion for 3 outcomes

I have been exploring the Kelly criterion for optimizing the bet size for a two outcome bet situation. I'm having trouble applying this to a three outcome bet. I may refer to this excellent thread: ...
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2answers
47 views

Parity of the sum of consecutive Bernoulli random variables

$\newcommand{\Var}{\operatorname{Var}}$I have $X_1,X_2,\ldots,X_{n+1}$ i.i.d. rv, each $X_i$ is a Bernoulli rv with parameter $p$, i.e. $X_i \in \{0,1\}$, $P(X_i=0)=1-p$ and $P(X_i=1)=p$ with $0 \leq ...
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4answers
73 views

Estimate bias of a coin

Consider a coin with probability $p$ of landing on head. You can estimate the prob by tossing it lots of times and looking at the proportion of heads one gets. In my problem I just want to know if ...
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2answers
79 views

How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
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0answers
28 views

Find one-dimensional distribution function $F(y\mid t)$ of random process $Y(t)$

$ Y(t)=tZ^2;\quad Z\sim U(-2;2); \quad t\ge0. \quad$ I need to 1) find one-dimensional distribution function $F(y|t)$ of random process $Y(t)$. 2) calculate probability that trajectory of the ...
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2answers
88 views

10 little dwarves

A dwarf-killing giant lines up 10 dwarfs from shortest to tallest. Each dwarf can see all the shortest dwarfs in front of him, but cannot see the dwarfs behind himself. The giant randomly puts a ...
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26 views

Probability of infinite intersections

While I was studying Probability and random processes I came across the following question. Say I have $A_1,A_2, \ldots, A_n$ events such that $A_i$ is in $E$ but not equal to $E$. What is: ...
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1answer
253 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...