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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10 views

Expectation of Gaussian Ratios

Consider the following expression: $z = \frac{\mathbf{x}^H P \mathbf{x}}{\mathbf{x}^H \mathbf{y}}$, where $\mathbf{y}$ is fixed (not random) and $\mathbf{x}$ is a complex Gaussian vector of zero mean ...
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12 views

Induction probability prove

Prove by induction $$P(\bigcup_{1\le i\le n}A_i)=\sum_{1\le i\le n}P(A_i)-\sum_{i_1<i_2}P(A_{i_1}\cdot A_{i_2})+\sum_{i_1<i_2<i_3}P(A_{i_1}\cdot A_{i_2}\cdot A_{i_3})-...+(-1)^{n-1}P(A_1\cdot ...
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1answer
24 views

Problem related to unbiased cubic dice

Consider an unbiased cubic dice with opposite faces coloured identically and each face colour red, blue or green, such that each colour appears only 2 times on the dice. If the dice is thrown thrice, ...
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0answers
8 views

Probability of an Athlete keeping their ranking

I am trying to figure out the probability that an athlete will keep his/her ranking in a specific sport for 2017. Obviously a basic probability function can show me the probability but I think the ...
1
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0answers
40 views

Multivariate to univariate distribution

Say one has a Student's t-copula (where all the margins and the copula can have different degrees of freedom). If you had a matrix of data (f.e. financial returns) and you know that the portfolio ...
6
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2answers
111 views

probability of sorted array with duplicate numbers

Suppose I have a sequence of n numbers {a1,a2,a3,...an} where some of the numbers are repeated. What is the probability that the sequence is sorted?
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6 views

Conditional Probability. Problem reconciling terms and definitions (sample space, outcome, event, experiment, trial, independence…)

Let's say I ask "What is the probability of getting a Heads given that you got a Tails?". In other words, I'm asking for $P(H|T)$. There are 2 interpretations: We're talking about a single coin ...
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0answers
8 views

Conditional probability of a random walk hits position $b$ in $n$ steps

This question comes from my question Modified gambler's ruin problem: quit when going bankruptcy or losing $k$ dollars in all Generally, I know the probability that a random walk hits position $b&...
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1answer
72 views

Modified gambler's ruin problem: quit when going bankruptcy or losing $k$ dollars in all

In each round, the gambler either wins and earns 1 dollar, or loses 1 dollar. The winning probability in each round is $p<1/2$. The gambler initially has $a$ dollars. He quits the game when he has ...
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2answers
14 views

Example of a set and two $\sigma$ algebras such that union is not a $\sigma$-algebra

What is an example of a set $X$ and two $\sigma$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, each consisting of subsets of $X$, such that $\mathcal{A}_1 \cup \mathcal{A}_2$ is not a $\sigma$-algebra?...
2
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0answers
17 views

Variance of sum of linear combination

I want to calculate the variance of a sum of linear combinations, so $$\operatorname{Var}\left(w'R_1 + w'R_2\right)$$ where $w$ is a $N\times 1$ vector and both $R_1$ and $R_2$ are $N\times 1$ ...
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0answers
10 views

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
6
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2answers
257 views

Variance of exit time for simple symmetric random walk

For a simple symmetric random walk starting at 0 (that is, a Markov chain on the integers starting at 0 with equal probabilities of going to the left and right at each step), I want to compute the ...
0
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3answers
291 views

Cover a line segment randomly with smaller line segments

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem to have been ...
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1answer
24 views

Computing the distribution function

I have troubles solving this task: Let $U_1,U_2,\dots$ be an i.i.d. sequence of random variables with uniform distribution on $[0,1]$. We set for every integer $n\geq 1$ $$M_n=\max\{1/\sqrt{U_1},\...
1
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2answers
56 views

Why does the normal distribution describe data collected in real life so well? [on hold]

$$ P(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( - \frac{(x-\mu)^2}{2\sigma^2} \right) $$ Is there any intuition behind choosing $e^{-x^2}$ instead of some other function?
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2answers
27 views

im getting confused in persentage question of probability..

a group consists of equal number of men and women of this group 20% of men and 50% of the women are unemployed. if a person is selected at random from this group the probability of the selected person ...
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1answer
21 views

Compute expected received balls from boxes

I have 6 boxes: $A,B,A',B',C \text{ and } D$. The box $A$ has $n_1$ red balls that are numbered from $1, \cdots, n_1$. The box $B$ has $n_2$ green balls that are numbered from $1, \cdots, n_2$. Make a ...
2
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2answers
17 views

Dash and dot probabilities

This is problem 1.41(b) in Casella and Berger's Statistical Inference. Consider telegraph signals "dot" and "dash" sent such that $$\mathbb{P}(\text{dot sent}) = \dfrac{3}{7}$$ and $$\mathbb{...
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0answers
9 views

Estimate CI for number of sock pairs in drawer

Here's my problem; I've described it terms of a sock drawer problem, but it actually has to do with something else (in this case surgeons doing concurrent surgeries) There are $N$ socks in a drawer. (...
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1answer
719 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
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84 views

Central Limit Theorem Exercise!

Let $X_1,...,X_{36} \overset{i.i.d}{\sim} \mbox{Poisson}(1)$ . Find $P( X< 1.1)$ by (a) using the central limit theorem of $\bar{X}$; (b) using the exact distribution of $\sum\limits_{i=1}^{36} ...
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0answers
14 views

Help with Boolean algebra

Consider a system with $n$ units where each unit is either working or failing. $x_j=1$ represents the condition that $j$-th unit is working. Suppose each unit is working with independent probability $...
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0answers
15 views

Determining the spherical coordinate parametrization of an area in $\mathbb{R}^3$.

Let $\epsilon_0,\epsilon_1,\epsilon_2$ be three iid random variables with a symmetric distribution and let $\lambda>1$. I want to calculate $$ P(\epsilon_0>0 \quad;\quad\epsilon_1>-\lambda\...
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0answers
22 views

What is probility to miss at least one test?

The probability that a teacher will give an unannounced test during any class is $\large \frac 15$. If a student is absent twice, then probability that he misses at least on test is $a) \ \...
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0answers
50 views

area of a rectangle

I know that to use polar coordinates instead of $dxdy$ we have $dA=rdrd\theta$. As such, we can have a double integral like $$ \int_{\theta=a}^{\theta=b}\int_{r=c(\theta)}^{r=d(\theta)}f(r\cos\theta,r\...
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0answers
15 views

Probability that the convex hull of random points is a triangle

Question: Given a fixed number $k > 3$ of random points in the plane, distributed according to a 2D standard normal distribution, what is the probability that all of them lie within the same ...
0
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1answer
16 views

Probability question with a radio competition

I'm quite new to statistics and I'm going through a few exam questions but I am a bit stuck on this one: ...
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2answers
31 views

Probability of incorrectly spelling a word

I'm currently trying to teach myself Statistics and have an exam question that I need a bit of help on: ...
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0answers
8 views

The coordinate difference of a nearest neighbor in a random set

Consider the following problem: $r$ vectors of length $t$ are drawn randomly, where each coordinate is an i.i.d Bernoulli random variable with success probability $$p_i , i=1...t.$$ What is the ...
8
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1answer
69 views

Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
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0answers
13 views

Fourier Analysis for Derandomization of Functions

I was wondering if there was an extension to Fourier Analysis on Boolean Functions. Specifically, it's well known that for any boolean function $$f: \{-1,1\}^{n} \rightarrow [-1,1] $$ we can decompose ...
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2answers
422 views

Health Risk Probability

Question: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman ...
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1answer
10 views

loss function similar to normal density

let $$L_\epsilon(x,p) = -\frac{1}{\sqrt{\epsilon}}\exp\left\{-\frac{(x-p)^2}{\epsilon}\right\}$$ be a loss function. given a random variable $X$ with density $f$ (possibly restricted), the risk ...
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0answers
16 views

Maximum value of $(1-F(t))t$ for probability distribution

Consider a continuous distribution on $(0,1)$ with cumulative distribution function $F$. For the value of $t\in(0,1)$ that maximizes $$P(t)=(1-F(t))t,$$ what is the lower bound of $P(t)$? For example, ...
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0answers
24 views

Conditional Expectation with linearity

Solving $E(X)$ $$=E(X-a+a)$$ (By linearity) $$=E(X-a)+E(a)$$ $$=E(X-a)+E(a)$$ $$=E(X-a)+a$$ Does this hold for all probability distributions? The place where this seems counter-intuitive to me is ...
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0answers
54 views

Ito's Formula applied to a weird equation…

EDIT: One thing I forgot to mention before is that this is all under the $\mathbb{Q}$ measure in case that changes anything I was just wondering if someone could explain how to solve this problem. I ...
4
votes
2answers
127 views

What is the expected value of $\min\{|X|,|Y|\}/\max\{|X|,|Y|\}$ assuming $X$ and $Y$ are independent?

So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent. What I am having trouble seeing is whether $\min\{|X|,|Y|\}$ and $\...
2
votes
3answers
24 views

Tricky permutations question

There are 8 buckets, each bucket is a different color (for simplicity, let's label the colors A, B, C, D, E, F, G and H; if you like: Aqua, Brown, Cyan, Diamond, Eggshell, Fuchsia, Green, Hot-pink). ...
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1answer
1k views

Conditional independence property: weak union

Let $(X,Y,W,Z)$ be disjoint sets of random variables each with finite space. Then prove that if $\Pr(X\mid W,Y \cup Z)=\Pr(X\mid W)$ then $\Pr(X\mid Y,Z \cup W) = \Pr(X\mid Z \cup W)$. This is ...
0
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1answer
43 views

Clarification of a Probability

Suppose I have two continuous, non-negative random variables, $X$ and $Y$ and I have that $$ P(X) = P(X|Y)\cdot P(Y). $$ Can I go on and say that $$ P(X\gt z) = P(X\gt z | Y\gt z)\cdot P(Y\gt z) = ...
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2answers
19 views

Conditional probability distribution and prior

In a linear Gaussian model, when I multiply a prior distribution $p(x)$ with the conditional $p(y|x)$ (here x and y are vectors), which one do I get: The joint distribution p(z) where z is the ...
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1answer
532 views

Conditional Probability teenage drivers

Teenage drivers pay more for automobile insurance than older drivers. Many companies offer discounts for teenage drivers good grades. Assume that 20% of all teenage drivers are involved in accidents ...
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0answers
23 views

MLE and unbiased estimator of $P\{X_{i}=1\}$ given poisson distribution

$\{X_{i}: 1\leq i \leq n\}$ is an i.i.d. Poisson random sample with unknown mean $\lambda$. Find the MLE of $P\{X_{i}=1\}$. Is the MLE unbiased? Does there exist an unbiased estimator of $P\{X_{i}=...
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0answers
9 views

Existance of a UMVUE

$ \{X_{i}: 1\leq i \leq n \} $ is a random sample, i.i.d $ N(\mu, 1) $ with $ \mu $ unknown. For a fixed $ x_{0} $, does there exist a UMVUE for $ \phi(x_{0}-\mu) $, where $ \phi $ denotes standard ...
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3answers
52 views

Expectation over sequencial random shuffles

I'm trying to understand this concept with this following problem: Logan is cleaning his apartment. In particular, he must sort his old favorite sequence, , of positive integers in ...
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3answers
38 views

Normal distribution

can anyone help me calculate $E(Z^4)$, $E(Z^3)$ for $Z\sim N(0,1)$? I know that $Z^2\sim \chi^2(1)$ then $E(Z^2)=1$, $Var(Z^2)=2$. Thank you.
2
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0answers
25 views

What is probability that a team reaches final if we know the probabilities of all opponents in the semi-final?

Our Discrete Math professor asked us a question as the Euros are going on. Given the following info, what is the probability that Portugal will make it to the final? Win Probabilities in quarter ...
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2answers
79 views

How to understand this integral result?

I was reading this page on Wikipedia: Birthday Attack I can understand up until how to approximate the minimal number of attempts for a given probability $$n(p; H) \approx \sqrt{2H \log \frac 1{1-p}...