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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0answers
41 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or more?

I have made a probability game, where you have to pull out 5 cards (from a deck of 52 cards and two jokers {54 cards total}), and if they add up to 40 or more, they win. Also, if the player pulls out ...
1
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0answers
33 views

proving a statement on Measure theory

Consider $(\Omega, U, \mu)$ be a measure space and X be an integrable function and for $A$, $\{A_n\}\in \mathscr{U};n\in \Bbb N$ I need to show that $\int_{A_n}X d\mu \to_{n\to \infty}\int_A Xd\mu$ ...
2
votes
1answer
35 views

What's the probability that the first four children born are boys and the last two children born are girls?

I'm having some problems with determining how to calculate a question about the gender proportion in newborns in some random family. A family consists of 6 children. The probability of a boy being ...
0
votes
1answer
10 views

Variable drawn from a normal distribution

What exactly is the meaning of a "variable drawn from a normal distribution"? I know what a normal distribution is, but my main exposure to "variables" is from calculus, so I have a hard time ...
1
vote
2answers
28 views

Odd Power terms of binomial theorem proof

I want to acquire all the terms of $(p+q)^n$ where the power of p is odd. Note that $p=1-q$ ($p$,$q$ probabilities) Ex. For $(p+q)^2=p^2+q^2+2pq$ I want to acquire only $2pq$(only term with odd ...
0
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0answers
18 views

Combining Sample Means and Variances

Let us assume we have several samples of unknown size but known mean value $\mu_{i}(x)$ and known variance $\sigma_{i}^{2}$. Now we want to calculate the mean value and variance for the total ...
2
votes
2answers
22 views

Symmetry in Probability Around a Particular Phenomenon in Time?

This has been hurting my brain substantially, recently. I'm not sure if I'm failing to make connections or if I see connections but am weary of their relevance. In my text the author claims that ...
0
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0answers
17 views

Question concerning invariant distribution

Let us consider the Markov chain $(X_n)_{n \in \mathbb{N}}$ with state space $I = \{0,1\}^m$ and transition probabilities $$ p_{xy} = \begin{cases} m^{-1} &\mbox{if } \vert x - y \vert = 1 \\ 0 ...
2
votes
2answers
37 views

Confidence Interval - Cigarette HW Question

Due to a lack of general student discussion on the message board my class uses, I've decided to ask this here. I want to know if I proceeded with this question correctly and if my choices were ...
-1
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0answers
38 views

Interesting and challenging problem [on hold]

I've been given this problem to solve, but didn't succeed until now. Can you help me? A city has 5 billion paper money (bills) in circulation. Thirty million of them are taken daily to the bank ...
1
vote
2answers
17 views

Probability of choosing two bulbs with the same rating given that one has a specific rating

I am trying to teach myself statistics, and working through Jay DeVore's excellent text of "Probability and Statistics for Engineering and the Sciences". The problem is as follows: We have box of the ...
0
votes
1answer
17 views

Independence - Probability and Statistics

Any help on this problem is greatly appreciated! I'm completely stuck School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A ...
0
votes
0answers
8 views

probability generating function moments for the multivariate case

Suppose ${\bf X} = (X_1, \ldots, X_d)$ is a non-negative integer-valued random vector, with pmf $p$, I tried to extend the results of the univariate generating function to the multivariate case, is ...
0
votes
2answers
32 views

proof of conditional probabilities

show that if the conditional probabilities exist then $$p(A_1\cap A_2 \cap \cdots \cap A_n) = p(A_1)p(A_2\mid A_1)p(A_3\mid A_1\cap A_2)\cdots p(A_n\mid A_1\cap A_2 \cap A_3\cap\cdots\cap A_{n-1})$$ ...
1
vote
1answer
20 views

How to make 4608 combinations with these choices? (Probability, permutations/combinations)

This problem has been giving me a lot of trouble... Freeze King claims to offer 4,608 different ice cream cups. A customer can choose from 3 sizes, 4 flavors; a waffle cone, sugar cone, or cup; ...
0
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0answers
4 views

p-average compound metric

I'm trying to prove that probability space metric defined as $d(X,Y)=(\mathbb{E}|X-Y|^p)^{1/p}$ is a metric indeed. Literature states that $d(X,Y)=0$ implies $Pr(X=Y)=1$, but no further explanations ...
1
vote
1answer
26 views

probability of 26 letters

A monkey at a typewriter types each if the 26 letters of the alphabet exactly once, the order being random. A. What is the probanility that the word HAMLET appears somewhere in the string if letters? ...
0
votes
1answer
19 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
0
votes
1answer
43 views

A measure theory question-1 [on hold]

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
0
votes
1answer
39 views

Probability of last cheese

I hope that someone could help me with understanding the exercise. In a cycle shaped house there are n chambers. In this house there is a mouse and each chamber has cheese except the room where the ...
0
votes
2answers
31 views

A probability theory question [on hold]

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
1
vote
3answers
50 views

Intuition behind independence result

The following problem is from Wasserman's $\textit{All Of Statistic}s$. I have worked through the algebra to arrive at the result, but it still seems very strange to me, so I would appreciate any ...
0
votes
0answers
8 views

Cardinality of maximum independent set for a given degree distribution

Consider undirected graph $G(V,E)$. Assume that $f_n(k)$ be the probability mass function of degree of a vertex in $G$. Further, assume that $f_n(k)$ is an strictly decreasing function of $k$ with ...
2
votes
1answer
24 views

Expected value - product of functions of uniformly distributed variables

We have $n$ random variables $X_1,...,X_n$, $n\geq 2$, where $X_i∼U(0,1)$ and all of them are iid. Let $ Z=\min(X_1,...,X_n)$ and $ \overline{X} = \frac{1}{n}\sum_{i=1}^{n}{X_i}$. Calculate ...
-1
votes
1answer
27 views

Lemme itô and Martingale [on hold]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
5
votes
1answer
20 views

simulating a fair six with a four equal sector spinner

Whist teaching basic probability I needed a group to use a fair four sector spinner but I'd none left. I gave them a die asking them to disregard 5,6 should they arise. The problem got me thinking ...
0
votes
1answer
40 views

proving a statement based on probability theory [on hold]

Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$
3
votes
1answer
41 views

Why this solution of the birthday problem is wrong? [duplicate]

If we have $n$ people there are $n(n-1)/2$ possible pairs that we can find. The probability that any two people have the same birthday is $1/365$. So for $n$ people the probability of finding at least ...
1
vote
3answers
20 views

Given the percentage, what's the probability it will happen exactly?

If a drug is effective $75\%$ of the time, what's the probability that it will be effective on EXACTLY $15$ out of $20$ people. Is there a formula or list of steps for this type of question?
0
votes
2answers
20 views

Can't find intersection of two probabilities.

I have the following problem: While producing goods, defect through event A has 3% probability and defect through event B has 4% probability. Total goods that are not defected - 95%. Find correlation ...
0
votes
1answer
11 views

Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
2
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0answers
29 views

Probabilistic Logic

I was wondering if there is any system of logic that has been worked out that explicitly uses probabilistic notions at its foundation. It would include ideas like as a first principle, all statements ...
0
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2answers
28 views

Determing a transition probability matrix

I need some support with this homework exercise: An urn contains at most $N$ balls. Let $X_n$ be the number of balls in the urn after the $n$-th execution of the following procedure: If the urn is not ...
0
votes
2answers
34 views

Integration limits of a Marginal Probability Density Function with a Triangle-Shaped Boundary

I have given a triangle shaped boundary $M$ of my probability density function in $\mathrm{R}^{2}$, with the limitations beeing: $$y = 0$$ $$y = x$$ $$y = 2-x$$ and a probability density function $$ ...
1
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0answers
21 views

Poincarè inequality in probability

I'm looking for a proof of the poincarré inequality in a probabilitic setting. That is to say, let $\mu$ be a probability on $\Bbb R^n$, what are the hypothesis in order to have, for any f smooth ...
0
votes
1answer
18 views

independence and characteristic functions [duplicate]

Why is it that \begin{equation*} \mathbf{E} [e^{i t_1 X_1} e^{i t_2 X_2}] =\mathbf{E} [e^{i t_1 X_1}]\mathbf{E} [e^{i t_2 X_2}] \end{equation*} for RVs $X_1, X_2$ and all $t_1, t_2\in\mathbb{R}$ ...
1
vote
1answer
21 views

probability density and distribution functions

I have $6$ independent and identically distributed variables such that $C_i \sim N(1000,400)$. 1) Calculate the density functions, distribution function and characteristic function of $C = ...
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0answers
20 views

Confidence interval of exponential random variables

I have a sequence of random variables $X_1, X_2, ..., X_n$ such that $X_i = e^{-(x_i-Θ)}$ I have to construct a confidence interval of the form $[Θ−c,Θ]$,where $Θ = \min _i{X_i}$. For $n = 10$ how ...
0
votes
0answers
7 views

Galton Watson process

Let $X_n$ the number of individus of the $n^{th}$ generation. For example suppose that a father has no brother and sister and does $3$ children. Suppose that thefather is the generation $0$ (i.e. ...
1
vote
2answers
29 views

What's the chance of $(\frac{1}{2})^x$ with $y$ iterations?

If I have a program that creates, let's say, one billion integers, with each having a pure $50 - 50$ chance to be one or zero, what is the chance of finding $x$ zeros in a row? for brownie points, ...
-4
votes
1answer
28 views

Four letters {A, B, C, D} are arranged in a line. What is the probability that A and B will be next to each other? [on hold]

Four letters $\{A, B, C, D\}$ are arranged, with no repetitions and always using the four. What is the probability that $A$ and $B$ will be next to each other?
-2
votes
0answers
15 views

Theoretical probability that everyone in the U.S. is separated by 6 degrees [on hold]

The six-degrees-of-separation theory says that I can be most certain that I have a friend who has a friend, who has a friend, who has a friend, who has a friend, who has a friend, who is friends with ...
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votes
0answers
15 views

Convercenge in probability implies convergence in Lp [on hold]

Show that if $X_n$ is that $|X_n|< C$, with $C\in \mathbb{R}$, $\forall n \in \mathbb{N}$, then $X_n \overset{P}{\rightarrow} 0 \implies X_n \overset{{L^P} }{\rightarrow}0$
0
votes
1answer
12 views

convergence of continuous mapped RVs

This is an extension of the result in my textbook, I'm wondering if it's true and if there are any references to it's proof. Let $X_n$ be a sequence of random vectors in $\mathbb{R}^d$, let $g : ...
0
votes
1answer
21 views

Confidence Interval Question - Steps Taken, no given standard deviation

I just wanted to make sure I was doing this Confidence Interval problem correctly (or incorrectly). Q: The following are the daily number of steps taken by a certain individual in 20 weekdays. (some ...
1
vote
2answers
60 views

Probability in a Restaurant

In a revolving restaurant, there are four round tables each with three seats. How many different ways can $12$ people sit in this restaurant? This is what I think the answer is: $$\binom{12}{4} ...
3
votes
2answers
63 views

What are the odds of any role of a 24 sided die occurring 4 or more times in 10 rolls?

Note that I am not asking about the odds of a chosen roll happening 4 times in 10 rolls, (this has a probability of 0.000517 according to a binomial calculator), rather, the odds of ANY roll happening ...
1
vote
1answer
18 views

Hypergeometric function variance

In a fishing event, a small lake is populated with $75$ trout, among which $25$ are tagged. Each participant is allowed to capture $5$ fish during the day (the fish are not put back into the lake). ...
1
vote
1answer
21 views

Two related question, in one. Same topic: Dispersion..

$1.$ Prove: If $X_1,X_2,X_3,\ldots,X_n$ are independent random variables then: $$D\left(\sum_{i=1}^n X_i\right)=\sum_{i=1}^n D(X_i)$$ Proof: Because of independence we have: $$D(\sum_{i=1}^n ...
0
votes
0answers
17 views

Proving that each element in reservoir have equal probability of been selected in reservoir sampling?

Here is the description of the algorithm and proof of the correctness The algorithm creates a "reservoir" array of size $k$ and populates it with the first $k$ items of $S$. It then iterates through ...