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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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1answer
11 views

proportion of the voters/ Central limit theorem

I want to compute the proportion of the voters p. Therefore I consider random variables $X_k$ for $k=1,...,n$: $$ X_k=\left\{\begin{array}{ll} 1, party \ is \ elected: "p" \\ 0, party \ ...
1
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1answer
17 views

Duration to guess password

I have been super stuck on this problem for a while and thought I turn to some expert help. My problem question: A password has length $8$ with a mix of $1$ uppercase letter (from $A$..$Z$), $5$ ...
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0answers
15 views

Proof by induction for a pdf

I am trying to solve this question, but I do not understand how to do induction on min(K,l). Can somebody explain this to me? Thank you!
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3answers
33 views

In determining probability using 2 dice rolls why are permutations (x,x) not counted twice?

So I've been working in probability regarding dice rolls. I came across this problem: If you roll 2 dice, what is the probability the first die is a 6 given that you rolled an 8? This is clearly a ...
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0answers
18 views

Regression on trivariate data with one coefficient 0

Suppose {$(x_i,y_i,z_i):i=1,2,...,n$} is a set of trivariate observations on three variables:$X,Y,Z$, where $z_i=0$ for $i=1,2,...,n-1$ and $z_n=1$.Suppose the least squares linear regression equation ...
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2answers
21 views

probability distribution proof $P(a < X \leq b) = F(b) - F(a)$

Let F be the distribution function of the probability $\mathbb{P}$ on $\mathbb{R}$ (induced by some random variable $X$). Prove: $\mathbb{P}((a,b]) =\mathbb{P}(a < X \leq b) = F(b) - F(a)$ This is ...
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1answer
1k views

If an ordinary six-sided die is thrown four times, find the probability of obtaining two even numbers and two odd numbers.

An ordinary die is has six faces numbered from 1 to 6. If an ordinary die is thrown four times, find the probability of obtaining two even numbers and two odd numbers.
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1answer
160 views
+300

If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d|x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: Use ...
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0answers
17 views

Solution Verification. How to find $P(X + Y \le tZ)$?

Let $X, Y, Z$ ~ $U(0, 1)$. For a fixed parameter $t$ find $P(X + Y \le tZ)$. I believe the problem requires handling different possible values of $t$. To begin with, I will assume $t$ is positive, ...
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0answers
16 views

Marginal density function comes out negative

I am having a problem understanding marginal functions in a class problem. The exercise says: Given the bidimensional density function $f(x,y)=6(3x-y)$ when $0<x<y\leq 1$ Find the probability ...
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0answers
24 views

Bernoulli's trial

In Bernoulli's trial with success probability $p$. Mean is (expected value) $p$. Expected value of first success should be $>1$. How can have success in less than $1$ trial which is $p$
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5answers
1k views

Expected number of tosses before you see a repeat.

Suppose we roll a fair die until some face has appeared twice. For instance, we might have a run of rolls 12545 or 636. How many rolls on average would we make? What if we roll until a face has ...
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1answer
28 views

Confused about the expression relating the CDF and expected value of a random variable

Let X be a random variable that takes on nonnegative values and has distribution function $F(x)$. $E(X) = \int_{0}^{\infty}1-F(x)dx$ The proof my book gives is: This statement was included at the ...
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2answers
916 views

Approximating Probability by Central limit theorem.

A large number of insects are expected to be attracted to a certain variety of rose plant. A commercial insecticide is advertised as being $99$%$ $ efective. Suppose $2000$ insects infest a rose ...
0
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1answer
20 views

Prove $2Cov(X_1,X_2)=E[(X-\bar{X_1})(X_2-\bar{X_2})]$

I'm trying to prove Hoffding's identity and I have trouble proving $2Cov(X_1,X_2)=E[(X-\bar{X_1})(X_2-\bar{X_2})]$. Where $\bar{X_1}$ and $\bar{X_2}$ are the independent copy of $X_1$ and $X_2$. Note $...
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1answer
882 views

How to calculate the average x-coordinate of a region?

I have a region in the x-y plane. Now I randomly select a point inside or on the boundary of the region. What is the expected value of the x-coordinate of my point? I.e. If I perform this operation ...
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1answer
21 views

show normality with non-linear transformation [duplicate]

This is one of the problem in the Allan Gut's Second Course for Probability. Let $X_1$, $X_2$ be independent standard gaussian e.g. N(0,1). Let $Y_1 = \frac{X_1^2 - X_2^2}{\sqrt{X_1^2 + X_2^2}}$, $...
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3answers
17 views

Given the bivariate standard normal density, how to show that $X$ and $Y$ are standard normal densities? [on hold]

Given the standard bivariate normal density with correlation coefficient $\rho$ for $X$ and $Y$: $$f_{X,Y}(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}e^{-(x^2-2\rho xy+y^2)/2(1-\rho^2)}$$ Is there a way to ...
0
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1answer
24 views

Finding the Probability of A

In a sample space, events $A$ and $B$ are such that $P(A) = P(B)$ and $P(A\cap B) = P(A'\cap B') $. What is $P(A)$? I came across this question. They pose it like there is only one correct answer is ...
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1answer
22 views

What is the probability that John and Michael sit next to each other at a round table? that they do not sit next to each other? [on hold]

John and Michael go to a cafe with $3$ friends. They get a round table with $5$ slots. a) What are the probability that John and Michael are sitting next to each other? b) Probability of NOT sitting ...
2
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1answer
18k views

conditional probability of an ace being drawn and then a king

I am trying to study for my statistic exam and not sure how to solve this question. The question is : what is the possibility of the following event: an ace is drawn first and a king is drawn ...
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1answer
11 views

Convergence to infinity of distinct partial sums

I am currently working what I thought would be a simple exercise, but am actually having trouble getting any foothold. For $n\in\mathbb{N}$ let $\{X_{n,k}\mid 1\leq k\leq n\}$ be i.i.d. random ...
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1answer
21 views

Use of conditional expectation in problem

A cricketer bats until he is out. For any given shot he takes, he either plays an attacking shot with probability $\frac{2}{3}$ or a defensive one with probability $\frac{1}{3}$. The type of each shot ...
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0answers
30 views

Probability and statistics: [on hold]

A uk distributor is to import 1000 mobile phones which will be sold with an insurance package for £400 each. They will be sent out to a number of shops in batches of 10. The distributor noted that, in ...
2
votes
1answer
37 views

Validity of Proof of Wald's identity

$\newcommand{\E}{\mathbb{E}}$ Theorem (Wald's identity): Suppose $\{X_i\}_{n \in \mathbb{N}}$ is a sequence of i.i.d random variables with $\E X_1 < \infty$. Let $\tau$ be a stopping time with ...
2
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1answer
62 views

Show that $X_n \to 0$ in probability under given condition.

Let $k > 0$. Suppose that $$\forall \epsilon > 0: \exists N: \forall n \geq N: P(|X_n| \geq \epsilon) \leq \epsilon k$$ Show that $X_n \xrightarrow{P}{ 0}$. Attempt: We have to show: $$\...
0
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0answers
18 views

Kolmogorov Complexity of a distribution

I read a paper that use a Kolmogorov complexity for a distribution, i.e., $K(p)$, where $p$ is a density function. However, I did not find the definition of it in this paper. Could anyone provides me ...
0
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3answers
41 views

How to calculate covariance with an minus like V(X-Y)?

my task is this: Be $ X $ and $ Z $ independent with the same distribution and $ Y :=X-Z . $ Calculate $ \operatorname{cov}(X, Y) $ and $ \operatorname{corr}(X, Y) . $ My Problem is the minus in $...
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1answer
27 views

Series Converging Almost Surely But Diverging in Mean

I am looking for an example of independent, non-negative random variables $X_1, X_2, \dots$ such that $$ \sum_{n=1}^{\infty} X_n \, \lt \, \infty $$ almost surely but $$ \sum_{n=1}^{\infty} \...
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0answers
39 views

Genius prediction [on hold]

It's a long story, but trust me, it's worth it. Matt is a detective and there's news of 3 murders about to happen in the upcoming week. Matt's usual success rate is 60%, i.e. He'll be successful in a ...
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2answers
19 views

Using the generating functions calculate the constant $C$.

Let $P (X=k, Y=j) = C \frac{ \binom{k}{j}}{2^{j+k}}$ , $k \in \mathbb{N}_{0}, 0\leq j \leq k$ be distribution of random vector $(X,Y)$. Using the generating functions calculate the constant $C$. I ...
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1answer
918 views

Probability between two numbers chosen from 1 to 30.

What is the probability that you choose an odd number or a multiple of six? The same number can be chosen twice. I was stuck on this problem earlier in class, and my teacher said the answer was 2/3, ...
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2answers
429 views

Is the chance of winning rock paper scissors $1/2$ or $1/3$? [on hold]

Is it $1/2$ or $1/3$. How do people think it is $1/2$? And what is the true answer, $1/2$ or $1/3$?
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0answers
18 views

Probability of choosing certain numbers [on hold]

What is the probability having the numbers 1,2,3,....,80 and choosing 20 of them to have exactly 10 odds and 10 even numbers.
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0answers
39 views

When do we have $E(p(Y|X)) = p(Y|E(X))$?

Let $p(Y)$ refer to the pdf of $Y$. I know that $\mathbb{E}_X(p(Y \mid X)) = p(Y)$. However, I am wondering if more can be said about the parameter of $p(Y)$ for parametric distributions. Are there ...
9
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5answers
555 views

How long until a random word with letters “A”, “B”, “C” ends in the pattern “ABC”?

Let's say I have word constructed from random letters, A Band C with $\mathbf{P}(A) = \...
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0answers
14 views

How do you calculate error bars on elo rating?

In season 14 of the Top Chess Engine Championship, Stockfish defeated Leela 50.5-49.5 in a 100-game match (the closest possible winning margin). Using an elo calculator such as this one, we can see ...
0
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1answer
28 views

Bayesian network problem: third day rainy, given first day is

I've tried searching for this problem online but could not find a solution, hopefully you can help me. I have three random variables [r1,r2,r3], these three variables shows the probabilities of it ...
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0answers
33 views

Gaussian process is a Brownian motion.

I am referencing from the book Probability from Dava Khoshnevisan. Thats my definiton of a brownian motion: 1) $W(0) =0$ and for all $t > 0$ is W(t) normal distributed with mean 0 and variance t. ...
0
votes
1answer
30 views

1st Yr Statistics: Hypothesis testing for the mean number of accidents at the 5 percent sig level

The following table gives the number of fatal accidents of U.S. commercial airline carriers in the $16$ years from 1980 to 1995. [See screenshot below] Do these data disprove, at the $5$ percent level ...
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1answer
20 views

Expectation of product of variables problem

A problem is asking to work out the total value of insurance claims made in a 400-day period. I am given that the expected value of any given claim is $£1000$. I am also given that the amount of ...
2
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1answer
35 views

Conditional expectation $E(X_1 \mid \overline{X}_n)$ if $X_1,\dots,X_n$ are i.i.d. Am I correct?

Conditional expectation $E(X_1 \mid \overline{X}_n)$ if $X_1,\dots,X_n$ are i.i.d. Since $X_1,\dots,X_n$ are i.i.d, then $E(X_1 \mid \overline{X}_n) = E(X_1)=\overline{X}_n$ Am I correct in thinking ...
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1answer
17 views

Distribution of $Y = b \bar{X}$ doesn't depend on $b$ (Gamma distribution)

Assume a random sample $X_1,X_2, \dots , X_n$ following the Gamma distribution with pdf $$ f(x) = \frac{b^a}{\Gamma (a)} x^{a-1} e^{-bx}, \, x>0 $$ ($a$ known, $b$ unknown). The distribution of $...
0
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1answer
32 views

How $\mathbb{P} (A | B \cup B^c)$ is $\mathbb{P} (A | B) \cdot P(B) + \mathbb{P} (A | B^c) \cdot \mathbb{P} (B^c)$?

P(A) = $P(A|\Omega)$         = $ P(A|B \cup B^c)$ But how to reach P(A | B) * P(B) + P(A | B$^c$) * P(B$^c$) from P(A | B U B$^c$) ?
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2answers
25 views

Inverse of sum of exponential random variables (mean and variance)

Assume $X_1,X_2, \dots , X_n$ following the exponential distribution with mean $\theta > 0 $ and the statistic: $$ T = \sum\limits_{i = 1}^n X_i $$ I know that the sum of exponential random ...
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1answer
22 views

Probability that the given students are not sitting adjacent to each other

Please note that I am not looking for a complete answer, but only hints on how to start. If you want to add a complete solution to help others who might want to know it, please put it in spoiler tags ...
0
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1answer
18 views

How is P(N|$H_3$) derived?

An exercise with a solution attached below. I do not understand how is, in ii), $P(N\mid H_3$) derived. Could somebody please explain me?
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1answer
10 views

A aperiodic state of a Markov chain has $N\geq 1$ such that $\forall n\geq N:p_{i,i}(n)>0$

The question I get asked is the following, I'm completely stuck on the problem: Let $i$ be an aperiodic state of a Markov Chain. Show that there exists $N\geq 1$ such that $p_{i,i}(n)>0$ for all ...
1
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1answer
1k views

Expected Value Question with Money

I've been struggling a lot w/ expected value, and was wondering if anyone could help me with this. I know the equation for expected value is $E(X)$ $=$ $NP$ The question asks: A 20 dollar bill, ...
0
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0answers
19 views

Closed-form solution to the problem: $\arg\min_{q \in \Delta_{\mathcal X} \text{ s.t } \|q-p\|_2 \le \epsilon} \mathbb E_q[X]$

Let $p$ be a probability distribution on countable set $\mathcal X$, denoted $p \in \mathcal X$. Question What is an "closed-form" solution for the problem: $\underset{q \in \Delta_{\mathcal X} \...