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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3 views

Why is this class closed under difference?

We have two independent random variables involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says in passing ...
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5 views

How to compute P(|X - E_Y[h(y)]| < c)?

Consider the discrete random variable $Y$, the continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, ...
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3answers
22 views

Events with States

Bob and I are playing a game with an unfair coin that is rigged to come up heads with probability $\frac35$ and tails with probability $\frac25$. Bob goes first, we take turns, and the first player ...
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0answers
21 views

vase with blue and red balls

At first I hope this is not a duplicate post. I tried to find it but I have not found it. I hope that someone could help me with understanding the exercise. This question is about a vase with r red ...
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1answer
29 views

Question about Measure Theory

Let $(\Omega, U, P)$ be a measure space and X be random variable and its distribution function $F_x(x)=P(\{\omega: X(\omega)\le x\})=P(-\infty , x]$ and the function F is continuous at x. If the ...
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1answer
26 views

interpreting wording of probability question

Two dice are rolled, and the sum of the face values is six. What is the probability that at least one ofnthe dice came up a three? I want to make sure that I am interpreting the language right when ...
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2answers
25 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
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1answer
16 views

What is the relation between$ P(A|B)$ and $P(A|B')$ for both independent and not independent events?

Let $A$ and $B$ be two events. If they are independent, how are $P(A|B)$ and $P(A'|B)$ related, if at all? If they are not independent, how are $P(A|B)$ and $P(A'|B)$ related, if at all? I've noticed ...
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18 views

Seven-Card Stud with Random Hand Selection

I was recently confronted with a number—$2727707$, actually—that started a short train of thought while I was placed on hold. (This seems to happen quite often: both the observation of unusual ...
3
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2answers
32 views

Conditional probability of exponential random variable

This question comes directly from a chapter in Gut's "Intermediate Probability" that focuses on conditional probability. I'm using this problem as more practice solving conditional probability ...
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3answers
69 views

Suppose a city with Three type of coins ?!

in a city we have tree type 1 dollar, 2 dollar, 3 dollar of coins. we want to pay for a 20 dollar product. how many ways we can pay for a 20 dollar product, if the seller has no money and number of 1 ...
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1answer
30 views

Marginal distribution of uniform distribution conditioned on poission??

Let $N$ be a Poisson distribution with parameter $\lambda$. Conditioned on $N$, let $X$ be uniformly distributed. What is the marginal distribution of $X$? ( this is one of my final exam problems ...
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1answer
27 views

Geometric Random Variables

I have a question that involves a certain criteria of a random variable as shown: The random variable $X$ has the distribution $Geo(0.2)$ and I would love it if someone could explain what the ...
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0answers
42 views

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

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 ...
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0answers
49 views

proving a statement on Measure theory [on hold]

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 $\int_{A_n}X d\mu \to_{n\to \infty}\int_A Xd\mu$ ...
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1answer
55 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 ...
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1answer
13 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 ...
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2answers
30 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 ...
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0answers
19 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 ...
3
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3answers
29 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 ...
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0answers
20 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 ...
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2answers
48 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 ...
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0answers
40 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 ...
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3answers
22 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 ...
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1answer
18 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 ...
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9 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 ...
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2answers
35 views

proof of conditional probabilities

show that if the conditional probabilities exist then $$p(A_1\cap A_2 \cap \cdots \cap A_n) = p(A_1)p(A_2\mid A_1)p(A_3\mid A_1\cap A_2)\cdots p(A_n\mid A_1\cap A_2 \cap A_3\cap\cdots\cap A_{n-1})$$ ...
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1answer
21 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; ...
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0answers
5 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 ...
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1answer
29 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? ...
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1answer
21 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 ...
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1answer
48 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 ...
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1answer
48 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 ...
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2answers
38 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) ...
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3answers
52 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 ...
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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
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1answer
29 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 ...
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1answer
31 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
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2answers
36 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 ...
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1answer
42 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
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1answer
42 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 ...
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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?
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2answers
21 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 ...
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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 ...
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31 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 ...
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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 ...
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2answers
35 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 $$ ...
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0answers
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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
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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 $t_1, t_2\in\mathbb{R}$ ...
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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 = ...