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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2
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1answer
303 views

Proof of Frechet-Hoeffding Copula bounds

How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$. I'm struggling to ...
0
votes
0answers
11 views

Show that $ \prod_{k=1}^n [1 + p_{nk}(e^{it} - 1)] \rightarrow e^{\lambda(e^{it} - 1)}, n \rightarrow \infty $

Suppose that $0\leq p_{nk} \leq 1, 1 \leq k \leq n$, $\max_{1 \leq k \leq n} p_{nk} \rightarrow 0, n \rightarrow \infty$ and $\sum_{k=1}^n p_{nk} \rightarrow \lambda$. Show that $$ \prod_{k=1}^n ...
0
votes
1answer
40 views

Total variation inequality

In the article from Lugosi and Devroye 1999 about testability of densities (page 13) I've stumbled upon this inequality: $$ V(f) \leq C(f)s(f) $$ where $f$ is a Lipschitz$(1)$ density with Lipschitz ...
1
vote
1answer
54 views

Calculate single “battle” outcome odds for RISK

I am trying to reproduce the values in this odds ratio table from Wikipedia. For all those unfamiliar with RISK, this is a game where units fight against each other via the roll of the dice: The ...
3
votes
1answer
16 views

Distribution of random variable

I need help with this problem: Let $(X_n)_{n\in \mathbb{N}}$ a sequence of i.i.d$\sim $Uniform$(\{0,\dots,9\})$ random variables. What is the distribution of $$X= \sum_{n=1}^{\infty} X_n 10^{-n}$$ ...
1
vote
1answer
842 views

probability of selecting cards

i'm confused by the following problem could someone walk me through it, so i can understand (a) 10 cards are drawn at random one at a time with replacement from an ordinary deck of cards. ...
1
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0answers
22 views

L1 error for scale/translation classes

This is an example given in the article about testability (Devroye and Lugosi 1990) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
1
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0answers
30 views

Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1999, page 4) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ). They use some properties of the ...
-1
votes
1answer
43 views

Combination Problem with Sofa [on hold]

Suppose we have 5 sofa on room A. in this room, 4 students seated on these sofa. These Strudents go to another room for eating dinner, and after that come back to room A. how many way the students can ...
0
votes
0answers
8 views

Generate quadrature points from a distribution

Is there any method to generate quadrature points from any arbitrary probability distribution, $p_{X}\left(x\right)$? We already know about Gauss Hermite rule for Normal distribution, Gauss-Laguerre ...
1
vote
1answer
18 views

Density of a sum

Let $X, Y$ be random variables having the joint density $f(x,y) = (a+1)(a+2)(y-x)^a, 0 \leq x < y \leq 1$, and $f(x,y) = 0$ elsewhere. Compute the density of $Z = X+Y$. My solution doesn't match ...
0
votes
1answer
25 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
0
votes
2answers
295 views

Poisson arrivals during an exponentially distributed interval

This is a marked homework question, so please try not to write complete solutions here: The number of customers that arrive at a service station during a time t is a Poisson random variable with ...
2
votes
1answer
22 views

Sufficient statistic

Let $\mathbf{X}=(X_1,\ldots,X_n)$ with joint frequency function $f(\mathbf{x};\theta_1,\theta_2)$ where $\theta_1,\theta_2$ vary independently. The set ...
0
votes
1answer
20 views

How to compute the difference between two numbers using a weighted formula?

Currently at work we use the formula ((expected-actual)/expected)*100 to show the difference between two numbers, however, this formula shows extreme differences for small numbers. For example, in the ...
3
votes
4answers
200 views

$\int_{0}^{\infty}xe^{-x^2/2}dx= 1$?

$X \sim N(0, 1)$ $$E(|X|) = \frac1{\sqrt{2\pi}}\int_{-\infty}^{\infty}|x|e^{-x^2/2}dx= \frac{2}{\sqrt{2\pi}}\int_{0}^{\infty}xe^{-x^2/2}dx=\sqrt{\frac{2}{\pi}}$$ I don't understand how the last ...
1
vote
1answer
33 views

Strange proposition in probability book for conditional probability

I found the following proposition (15.1) in the probability book of Heinz Bauer: Let us given that $X$ is a numeric random variable on $(\Omega,\mathcal{A},P)$ which is non-negative / ...
2
votes
2answers
73 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
0
votes
0answers
27 views

the usage of the PDF

We know that in a continuous distribution,the probability in a specific point like x is zero.If it is this way, then why does exactly the probability function-f(x)- of a distribution like normal ...
2
votes
3answers
1k views

Notation of random variables

I am really confused about capitalization of variable names in statistics. When should a random variable be presented by uppercase letter, and when lower case? For a probability $P(X \leq x)$, what ...
1
vote
1answer
21 views

Is it possible that a randomized recursion has a nonzero probability of either converging or diverging?

I have very little "hands-on" experience with probability, but here is my context: I was looking at the random Fibonacci sequence: $$f_0=f_1=1, f_n=f_{n-1}+Xf_{n-2}$$ where $X$ is chosen randomly ...
1
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0answers
19 views

Find joint distribution of distance to origin from points chosen uniformely inside spheres

I have the following problem. A point $P $ is chosen uniformely from the unit sphere $|X |\le1$, where $X \in R^3 $. Then an other point $P' $ is chosen uniformely from the spere centered at the ...
1
vote
1answer
59 views

Will I will be able to sit and watch the movie?

Recently I went to the theater. When I came to buy my $3$ tickets (two friends and I), the machine tells me that there is $18$ seats out of $300$ ($15$ rows of $20$ seats). What is the probability ...
3
votes
1answer
36 views

In 30 boxes are 15 balls. Chance all balls in 10 or less boxes?

Question1: I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i found 0 balls. Afer i collected all 15 balls i put them randomly inside the boxes. How much is the chance that all balls are ...
1
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2answers
27 views

(What is the formula to find) What is the probability that the sum of the numbers on the tickets chosen is at least 7?

Senario: Box A contains four equal-sized tickets, numbered 1, 2, 3 ,4 Box B contains three tickets of the same size, numbered 4, 5, 6 An experiment consists of selecting one ticket from the box A ...
0
votes
0answers
6 views

How to find confidence interval of quotient

This site does a good job of explaining how to find a confidence interval for reliability $R(t)$. It also explains how to find conditional reliability $R(t|T) = R(t + T) / R(T)$. However, it does ...
2
votes
1answer
21 views

What are the probability that the first two rows of the class are full?

I was boring in my class. So I ask myself the question: What are the probability that the first two rows of the class are full? Knowing that we're $25$ students in my class and the class have ...
2
votes
1answer
39 views

St. Petersburg and the law of large numbers

Recently I learned about the discussion around the St. Petersburg paradox and how people try to explain why the calculated expected value differs so much from most people's intuition. My question: ...
2
votes
2answers
33 views

$\overline{\theta}$ the maximum likelihood estimator of $\theta \implies$?

I can't understand how the following statement holds without any extra conditions on the function $g$: $\overline{\lambda}$ the maximum likelihood estimator of parameter $\lambda \implies ...
0
votes
1answer
15 views

How to solve disjoint probability problem?

Suppose the two events “high” and “low” make a disjoint partition of a sample space and “favourable” is any event. If P(high) = 0.3, P(low) = 0.7, P(favourable| high) = 0.9 and P(unfavorable| low) = ...
1
vote
1answer
30 views

Repeated Random Number Selection Until No Collisions - Number of Repetitions

I have a problem that has to do with my research that troubles me for a while and I could not find a solution either by myself or online. Suppose you have $n$ people in one room. Everyone picks a ...
0
votes
1answer
19 views

Probability to get in two ways

There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color? The ...
1
vote
1answer
504 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 $Pr(X|W,Y \cup Z)=Pr(X|W)$ then $Pr(X|Y,Z \cup W)=Pr(X|Z \cup W)$. This is sometimes referred to as weak ...
-2
votes
1answer
36 views

Combination Problem on Distinct Object [on hold]

How many way we can devide 10 distinct object between 5 student? i think the solution is combination(10,5)*5! but i'm not sure. or maybe 5^10? any expert could help me ? add some detail?
1
vote
3answers
53 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
1
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0answers
41 views

The probability that exactly / at-least $k$ numbers are in the correct position [duplicate]

Given a sequence of $[1,\dots,n]$ in random order: Let $P_k$ be the probability that exactly $k$ numbers are in the correct position Let $Q_k$ be the probability that at least $k$ numbers are in the ...
-3
votes
1answer
34 views

Non-Negative Integer answer of Inequation

i study some combination note. in setion inequation non-negative answer i have some misunderetsanding. suppose we have $7<=x_1+x_2+x_3<=12$, i want to calculate numbwe of Non-Negative Integer ...
0
votes
0answers
13 views

Relation between RSD and Mean

Is there any relation between relative standard deviation and mean? In my data there is 3 mean values like 98.1,99.5,97.8 and relative standard deviation is 0.6,0.7,0.8 respectively. Which one is ...
3
votes
1answer
36 views

Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
-1
votes
1answer
20 views

Probability question, statistics and actuarial science.

A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury ...
3
votes
2answers
35 views

Binomial dependent on a Poisson

I have been working on a problem with a binomial rv dependent on a poisson rv and have worked through to this point: $P(X=x) = \sum_{n=x}^{\infty} \dfrac{n!}{x!(n-x)!} p^x(1−p)^{n−x} ...
0
votes
3answers
35 views

What is the probability that a man appearing near the tower will stay in the dark for at least 5 seconds?

A searchlight on top of the watch-tower makes 3 revolutions per minute. What is the probability that a man appearing near the tower will stay in the dark for at least 5 seconds? The problem ...
0
votes
1answer
358 views

Probability Question (Possibly Bayes Theorem?)

I have a quick probability question. I've solved half of this problem and I'm not sure if this requires Bayes Theorem. Here's the question: "Suppose that a polygraph can detect 53% of lies, but ...
6
votes
3answers
151 views

Paradoxical Game Show Problem [duplicate]

Here's a problem that has had me scratching my head for a long time: Imagine you're in a game show, and are presented with 2 boxes. You are told that both boxes contain a sum of cash, but one of the ...
4
votes
1answer
54 views

Simple counting problem

Suppose that you have a box with $n$ balls, from the $n$ balls $k$ are white and $n-k$ are black. Now, sequentially you draw (without replacement) the $n$ balls in groups of $m$ (a natural number that ...
0
votes
5answers
45 views

What is the probability that Brian makes money on his first roll

Brian plays a game in which two fair, six-sided dice are rolled simultaneously. For each die, an even number means that he wins that amount of money and an odd number means that he loses that ...
4
votes
1answer
74 views

What is the probability that both roots of the equation $Ax^2 + Bx + C = 0$ are real?

Given this problem as part of prep for a test. We've done the same problem without A being a random variable, but I am completely stumped as to how to accomplish this one with three r.v.s I know the ...
0
votes
0answers
18 views

How to estimate $\sum_{x=1:n}{xf(x)}$ having $\tilde{f}$

I have an estimator $\tilde{f}(x)$ whose error is at most $\epsilon$, i.e., $\frac{|f(x)-\tilde{f}(x)|}{|f(x)|} \leq \epsilon$. I want to estimate $\sum_{i=1:n}i.f(i)$ with a small error. But if I ...
1
vote
2answers
27 views

Probability with Uniform Distribution with Multiple Variables

Every time you go to a beach for vacation, you take home a little sand to keep as a souvenir. Over your lifetime, you have done this exactly 100 times. On each visit, the weight of sand you take home ...
1
vote
1answer
20 views

Probability with Exp distribution, CDF, and multiple variables

You have a list of chores to do at home, but are expecting family to arrive shortly. The amount of time until their arrival (measured in hours) can be modeled as an Exp(2) random variable. Your list ...