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
153 views

Probability of Top Gear's alleged insulting number plate - requiring confirmation please [closed]

Please confirm or alter the calculations below which I believe to be correct based on 650 regional letter pairs for the 2nd and 3rd letters of the grouped three This UK number plate in question is ...
2
votes
1answer
207 views

Probability of selecting the winning numbers in a lottery

I've been studying combinatorics for a while. I've solved a problem but I'm not sure if I'm right. I'll just copy-paste the problem here. In a lottery, six distinct numbers are selected at random ...
2
votes
1answer
73 views

Joint distribution from conditional

So the question is: Let $\theta\sim U(0,1)$ and $X\mid\theta\sim\text{Binomial}(2,\theta)$. Find the joint distribution of $(X,\theta) $ The way the lecturer started was the following: $$ P(X=k, ...
1
vote
1answer
182 views

Solving probability question using the inclusion exclusion principle

Question: $F$ is the set of all functions from a $n$-set to $B=\{a,b,c\}$. In a random experiment of selection of functions from $f$ assume that every $f\in F$ is equally likely. What is the ...
2
votes
1answer
41 views

Two conditional probability questions about three chests with two balls each

Consider 3 chests. Each chest has 2 balls. Chest A has $\color{red}{both\,\,red}$. Chest B $\color{red}{one\,\,red}$ and $\color{blue}{one\,\,blue}$. Chest C has ...
0
votes
1answer
114 views

Let $X$ and $Y$ be i.i.d. $\operatorname{Geom}(p)$, and $N = X + Y$. Find the joint PMF of $X, Y, N$

Let $X$ and $Y$ be i.i.d. $\operatorname{Geom}(p)$, and $N = X + Y$. Find the joint PMF of $X, Y, N$. I have generally difficulties with such problems, as I get easily confused. Below I detailed ...
1
vote
2answers
70 views

finite variance but infinite higher moments

Is it possible to find a positive random variable with finite variance such as \begin{equation*} \mathbb{E}(X^{2+\varepsilon})=+\infty \end{equation*} for all $\varepsilon > 0$ ? Equivalently, is ...
7
votes
1answer
205 views

Puzzle: How Many Possibilities Are There Between Connected Points?

Puzzle Jenny drew on her page six points, as shown below: Jenny wants to build a cool match of her points. In a match , divide the six-point into pairs, so that each point has one partner exactly. ...
0
votes
1answer
192 views

Five cards are chosen from a standard deck with replacement. Find joint PMF of number of queens, kings and others.

Five cards are randomly chosen from a standard deck, one at a time with replacement. Let $X$, $Y$, $Z$ be the numbers of chosen queens, kings, and other cards. (a) Find the joint PMF of X, Y, ...
1
vote
1answer
103 views

Estimate the number of trials needed to observe all the possible outcomes of an experiment [duplicate]

I am stuck with the following problem: Each package of Pokemon cards contains 1 of N possible legendary Pokemon. How many packs do you expect to buy to get all N? We assume all N legendary cards are ...
1
vote
1answer
682 views

Expectation of hitting time for simple symmetric random walk

Assume there is a simple symmetric random walk $$S_n=X_1+...+X_n,\quad S_0=0$$ where $\mathbb P(X_i=\pm 1)=\frac{1}{2}$. Define $T=\inf\{n:S_n=1\}$. How to compute $\mathbb E(T)$? My idea: if ...
0
votes
1answer
82 views

Joint PMF of the number of men and women in a committee

A committee of size $k$ is chosen from a group of $n$ women and $m$ men. All possible committees of size $k$ are equally likely. Let $X$ and $Y$ be the numbers of women and men on the committee, ...
9
votes
1answer
176 views

Is Entropy = Information circular or trivial?

I have seen several "maximum entropy distributions" used in the mathematical and statistical literature, often with the justification that they are "minimally informed" beyond the assumptions and data ...
0
votes
3answers
47 views

If $X \sim Bin(n,p)$, using $E(X(X-1)) = g''(1) = n(n-1)p^2$ show that Var$(X) = npg$.

If $X \sim Bin(n,p)$ using $E(X(X-1)) = g''(1) = n(n-1)p^2$ show that Var$(X) = npg$. I understand that g is the generating function $g_{x}(t) = \sum_{k=0}^{\infty} p_{k}t^{k}$, and I know that the ...
4
votes
1answer
193 views

Probability that rolling X dice with Y sides and summing the highest Z values is above some value k

Some background: There is an RPG called Legend of the Five Rings, with an interesting dice system. You roll X dice, and keep the highest Z of them. You add those Z dice together. This is phrased as ...
0
votes
1answer
186 views

How write down joint PMF, conditinals and marginals of Bernoulli and Binomial random variables?

I have the following problem, where I'd like to know whether I'm doing it right and whether the notation of the joint PMF is correct. One of two doctors, Dr. Hibbert and Dr. Nick, is called upon ...
2
votes
0answers
87 views

Shannon Entropy, prove $H(Wx)=H(x)+\log|\det W|$

I'm doing an essay on ICA (independent component analysis), and I could use some help. In essence, ICA is an algorithm that minimizes the entropy of $n$ $1$-dimensional random variables, but to show ...
1
vote
2answers
47 views

Odds of outcome at least once over N repeated observations

A friend suggested that if the odds of something happening to you in one year is P, the probability of it happening over N years ...
1
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2answers
47 views

Probability of a subset of the first $n$ natural numbers are prime.

Suppose we have $P(x) = \frac{1}{n}$ for $x \in \{1,...,n\}.$ Is there a way to find out what the probability is for $P(A_p)$ where $A_p$ is the set of integers $x\in \{1,...,n\}$ such that $x$ is ...
8
votes
2answers
215 views

A man who lies a fourth of the time throws a die and says it is a six. What is the probability it is actually a six?

First, I apologise for the vague title. I couldn't think of a short way to represent the problem. Also, I am aware that a similar question exists, but I have a little bit more insight. The problem ...
2
votes
2answers
49 views

Is expectation of random variable independent of its characteristic function?

For any random variable, does that equation hold? I proved for normal distribution, but I can't generalize. E$[xe^{itx}] = E[x]E[e^{itx}]$ Thanks in advance.
2
votes
2answers
97 views

For every $\epsilon>0$, the probability of $W_t>(1+\epsilon)\sqrt{t\log(t)}$ tends to $0$ as $t\to\infty$

Can anybody give a hint to show for all $\epsilon>0$ $$\lim_{t \to \infty} P \left( \frac{W_t}{\sqrt{t\log(t)}}>1+\epsilon \right) = 0$$ with $W_t$ Brownian Motion? (Or W(t), a Brownian motion ...
1
vote
1answer
47 views

How many messages with exponentially distributed time can be sent with probability $\frac 1 2$ in at most $15$ minutes?

Assume that the time to send a message follows an exponential distribution with $\lambda = 8$ and is independent from the rest of the messages. The messages are sent sequentially, one after another. ...
1
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0answers
39 views

Probability of Team A shooting a given percentage from 3-point range

I would like to find a formula I can use to determine the probability of Team A shooting a specific percentage from 3-point range against Team B given Team A's overall average as well as the ...
1
vote
2answers
56 views

A decimal die is tossed 4 times, thus producing a series of 4 random digits…?

A decimal die is tossed 4 times, thus producing a series of 4 random digits. The series is said to be increasing if from the second digit onward, each digit stands for a larger number than the ...
3
votes
3answers
195 views

Uniform probability distributions

I have a bit of a misunderstanding going on: If we have a continuous random variable $x$ which is uniformly distributed between $0$ and $2$, then its probability density function is $P(x)=\frac{1}2$. ...
1
vote
1answer
176 views

Variance on the number of steps in an absorbing Markov chain

I've been looking for a proof for the variance on the number of steps before being absorbed in an absorbing Markov chain. The theorem is given on Wikipedia without citation. Following the references ...
1
vote
2answers
32 views

How to compute the expected value of the sum of two random variables?

How do you compute expected value? I need help for not word problems, but a question similar to this: Two numbers, $x$ and $y$, are randomly chosen from the interval $[0,1]$. What is the expected ...
1
vote
1answer
46 views

First time markov process spends tau units in certain state

Consider a continuous time Markov process $\{X(t)\}_{t≥0}$ on the state space $\{0, 1, 2, . . .\}$ with stationary probabilities $\{π_0, π_1, π_2, . . .\}$. Suppose that, when currently in state $i$, ...
1
vote
0answers
21 views

A problem on Ito integral [duplicate]

Let $W$ be a standard, one-dimensional Brownian motion. Let $T\in(0,+\infty)$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}\left|e^{-\beta t}\int_0^te^{\beta ...
0
votes
1answer
105 views

Likelihood Ratio and Neyman-Pearson factorization theorem

I'm looking at a family of distributions given by $P = \{P_{\theta} \quad | \quad \theta \in \{0,1\} \}$. I'm trying to prove that $$T(x) = \frac{p_{1}(x)}{p_{0}(x)}$$ (i. e. the likelihood ratio) ...
0
votes
4answers
55 views

the coefficient of $t^{18}$ in $t^6(1+t+\cdots+t^5)^6 $ equals the coefficient of $t^{12}$ in $(1+t+\cdots+t^5)^6 $.

Let $X$ be the total from rolling 6 fair dice, and let $X_1,\ldots,X_6$ be the individual rolls. What is $P(X=18)$? Then the PGF of $X$ is $$E(t^X) = E(t^{X_1} \cdots t^{X_6}) = \frac{t^6}{6^6}(1+t+ ...
3
votes
1answer
228 views

What is a mathematically rigorous justification for multiplying edge probabilities of a tree diagram

I was trying to understand why it was mathematically justified to multiply edge probabilities in a tree diagra and I came across the following question: Why do we multiply in tree diagrams? The ...
0
votes
2answers
560 views

probability - An urn contains 3 white balls and 4 black balls. Second urn contains 6 white balls and 4 black balls.

An urn contains 3 white balls and 4 black balls. Second urn contains 6 white balls and 4 black balls. From the first urn are draws 2 balls and they dropped in the second urn. Then from the second urn ...
1
vote
1answer
80 views

Does this variation of Jensen's inequality hold?

The original Jensen's inequality in probability theory is generally stated in the following form: if $X$ is a random variable and $f$ is a convex function, then $f \left(\mathbb{E}[X]\right) \leq ...
1
vote
1answer
59 views

For a pdf $f(x)$, how can we prove that $\int_{-\infty}^{\infty} x\,f(x)\,dx=\int_{-\infty}^{\infty} F(x\geq t)\,dt$?

$f(x)$ is a probability density function and $F(x)$ is the corresponding cumulative distribution function, i.e., we have the relationship on the derivative $\frac{d}{dx}F(x)=f(x)$. Given this, how ...
0
votes
1answer
44 views

A question about Bayesian Networks from Judea Pearl's book.

"Given a probability distribution $P(x_1, \dots, x_n)$ and any ordering d of the variables, the DAG(directed acyclic graph) created by designating as parents of $X_i$ any minimal set П$_{X_i}$ of ...
2
votes
1answer
145 views

3 dimensional $6\times 6\times 6$ lit cube problem involving looking for a specific lit pattern and quantity of them.

Suppose we have a $6\times 6\times 6$ cube such that it has $216$ subcubes, each with a visible, discernible light in it. A random number generator is connected to the cube and it will choose ...
2
votes
4answers
125 views

Intuition for regression to the norm

So for regression to the norm it says if someone has a high score on a test (relative to the average) then they are likely to score lower and lower on each following test? This seems very counter ...
4
votes
0answers
66 views

Interchangeability of the malliavin derivative with a lebesgue integral

I was curious to know the most general conditions under which a malliavin derivative $\mathscr{D}_t \int^T_t F_v d\mu(v) = \int^T_t \mathscr{D}_t F_v d\mu(v)$ commutes with a lebesgue integral? I was ...
1
vote
0answers
95 views

Explosion of a Markov chain

I am considering a Markov chain i continunous time on the countable state space $S=\lbrace 0 \rbrace\cup \lbrace (i,j) \mid i \in \lbrace A,B,C,D,E,F \rbrace , j \in \mathbb{N} \rbrace$. The ...
0
votes
2answers
133 views

Poisson process: Probability that $n$ busses of bus route A arrive before the first bus of route B

There are two bus routes that stop at a bus stop. Busses arrive according to a poisson process with rate $\lambda_1t$ buses per minute for the first route and $\lambda_2t$ for the second route. ...
1
vote
2answers
247 views

Probability in a Dice Game (Zombie Dice)

In the game of Zombie Dice (Rules) there exist 13 dice: 6 Green - 3 Brains, 2 Footprints, 1 Shotgun 4 Yellow - 2 Brains, 2 Footprints, 2 Shotguns 3 Red - 1 Brain , 2 Footprints, 3 Shotguns A ...
0
votes
3answers
125 views

If you attempt to predict a Roulette wheel $n$ times, what's the probability you'll get $5$ in a row at some point?

I'm talking about a Roulette wheel with $38$ equally probable outcomes. Someone mentioned that he guessed the correct number five times in a row, and said that this was surprising because the ...
4
votes
1answer
176 views

Probability of royal flush dealt to table of n players

I'm interested in calculating the probability of a royal flush being dealt to ANY of the n players seated at a Texas Hold'em poker game (2 hole cards, 5 community cards). The probability of YOU being ...
3
votes
1answer
526 views

Finding variance of the sample mean of a random sample of size n without replacement from finite population of size N.

I encountered this problem in the book "Introduction to the Theory of Statistics" (by Mood, Graybill and Boes) and I have not been able to solve part (c): "A bowl contains five chips numbered from 1 ...
4
votes
2answers
176 views

better expression for simple random walk

Let $P_{k,j}$ be the probability that a simple symmetric random walk starting from the origin reaches the point $k \in \mathbb{N}$ precisely in $j$ steps without ever returning to the origin. ...
1
vote
1answer
180 views

How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
0
votes
2answers
79 views

What is the probability of a chain of a given length in a random graph?

Let $G$ be an undirected graph with $n$ nodes. An edge is randomly and independently drawn from each node to any of the other nodes. If some arbitrary node $a$ is chosen, what is the probability that ...
1
vote
3answers
57 views

If a word appears with probability $0.05$, how many words are needed so that it appears with probability $0.99$?

Probability of a specific word appearing in a language is $0.05$. How many words must there be in a text, so that the word appears at least once with a probability of $0.99$? My understanding is ...