Questions tagged [probability]
For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].
106,102
questions
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5 consecutive heads in 25 coin tosses with linearity of expectation
I am reasoning around the linearity of expectation.
If for example, I want to know the expected number of a pair HH in 25 tosses (note that HHH has 2 possible HH pairs) I could use linearity of ...
- 560
2
votes
2
answers
887
views
If P(A|B)=1, then is it correct that P(A|B,C)=1 under the assumption that P(B,C)>0?
If $P(A\mid B)=1$, then what can we say about $P(A\mid B\cap C)$? Is it 1 or not?
The condition says that if B occurs, then A occurs a.s.
Then, if B and C occur, this implies that B has occurred and ...
- 466
2
votes
1
answer
84
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2 friends shooting
You and a friend play a shooting game. The chances of you hitting 1st and 2nd shot are 0.3 and 0.5 respectively. The chances for your friend are 0.4 and 0.4 respectively. Who has a better probability ...
- 560
2
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0
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141
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martingale small ball condition
I am reading this paper which defines a martingale small ball condition that I have never encountered in a standard probability book.
The definition is as follows (page 8, definition 2.1): Let $Z_t$, $...
- 307
2
votes
0
answers
559
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Convex combination of random variables
Let $X, Y$ two independent real random variables. If I define the distance of the distribution (i.e. the cumulative density function) of $X$ with a target distribution $F$ as
$$\sup_t |F_X(t)-F(t)|$$
...
- 1,149
2
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0
answers
52
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The probability of the matrix $A = (a_{ij})_{i,j\leq n}$ for $a_{i,j}\sim \text{Bernoulli}(\rho)$ is invertible
I have the next problem:
Let $A=(a_{i,j})_{1\leq i,j\leq n}\in\mathcal{M}_{n,n}(\{0,1\})$ such that $a_{i,j}\sim \text{Bernoulli}(\rho)$ for all $i,j=1,\dots,n$. How can I calculate the probability ...
2
votes
0
answers
98
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Probability of X lying within one standard deviation for different distributions?
We know that for the normal distribution (denote this $P_N(x)$), the probability of a random variable having a value within one standard deviation of the mean is approximately $68$ %. That is
$$\int_{\...
- 153
2
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0
answers
44
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Geodesics in distribution space with Jensen-Shannon metric
Suppose, $D$ is the space of all continuous distributions over $\mathbb{R}$, equipped with Jensen-Shannon metric:
$$d(P, Q) = \sqrt{\int_{-\infty}^\infty (p(t)\ln(1 + \frac{p(t)-q(t)}{p(t)+q(t)}) + q(...
- 15.6k
2
votes
1
answer
131
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Interpretation of standard deviation
I've come across many articles that describe standard deviation to be a measure of how much spread out our distribution is. In other words, it is a measure of how far away from the mean, we expect a ...
2
votes
2
answers
200
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Probability stick I drop parallel to diagonal of rectangle fits within the rectangle
Here's a question from my probability textbook:
A floor is paved with rectangular bricks each $a$ inches long and $b$ inches wide. A stick $c$ inches long is thrown upon the floor so as to fall ...
- 1,415
2
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0
answers
44
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Using mappings and bijections for a simple looking probability problem.
I came across this simple problem today: Show that the probability of rolling a $14$ is the same whether we use $3$ or $5$ fair dice. It is very easy (but tedious) to solve with basic probability ...
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2
votes
1
answer
3k
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An airline knows that 5% of the people who make reservations for a given flight do not show up.
An airline knows that 5% of the people who make reservations for a given flight do not show up. Consequently, its policy is to sell 52 tickets for a flight that can accommodate 50 people. What is the ...
- 23
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57
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Convergence of expectation of sequence of random variables
I have the following problem:
Let $Q_n$ denote a sequence of random variables such that
$\mathbb{E}[\frac{Q_n}{n}] \xrightarrow{n \rightarrow \infty} 0$
and
$\frac{Q_n}{n} \xrightarrow{n \rightarrow \...
- 41
2
votes
0
answers
40
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$p_{ii}^{(2n)}=1$ and $p_{ii}^{(2n+1)}=0$
I learnt the concept of the period of state i for DTMC and I'm wondering
why we have $p_{ii}^{(2n)}=1$ and $p_{ii}^{(2n+1)}=0$?
if $d(i)=p$ then $p_{ii}^{(pn)}=1$ and $p_{ii}^{(pn+1)}=0$?
- 81
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74
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Most likely number of people to be ill in parish of $n$ people
Here's a question from my probability textbook:
If the chance of any given person being ill at any given time is $\theta$, what is the most likely number of persons to be ill at once in a parish of ...
- 1,415
2
votes
0
answers
142
views
Probability measure notation
Has anybody seen probability theory combined with tensor analysis? I think they use a different kind of notation with the standard probability theory. For example in standard probability theory we ...
- 429
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votes
0
answers
2k
views
I have a square, and place three dots along the 4 edges at random. What is the probability that the dots lie on distinct edges?
The correct answer is listed as (3/4) * (1/2) = 3/8. The reasoning is because it doesn't matter where you put the first dot, but after you put the first dot, it is 3/4 chance of a different edge, and ...
- 51
2
votes
0
answers
258
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Cramers theorem for large deviations
Let $(X_n)$ be real-valued i.i.d. random variables such that the cumulant generating function $\Lambda(t):=\log E e^{tX_1}$ is finite for all t and let $S_n:=\frac{1}{n}(X_1+...+X_n)$ denote the ...
user941567
2
votes
1
answer
805
views
Understanding the confidence interval and statistical significance
I am struggling to understand confidence intervals and their relationships to a null hypothesis.
The basic definition of the confidence interval is: (1−α), where α is the statistical significance.
So ...
- 31
2
votes
0
answers
141
views
Probability that all the $n$ letters are not placed in the right envelope
There are $n$ letters and $n$ addressed envelopes. If the letters are placed in the envelopes at random, what is the probability that all the letters are not placed in the right envelope?
The number ...
- 4,055
2
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133
views
multiplication principle and permutation rule; when to use what
Problem: You have CDs of which n1= # of classical, n2= # of rocks, n3= # of pop. How many ways can we place the CDs on a rack but keep the genres together?
The Answer is:
Because there are n1 ways to ...
- 29
2
votes
0
answers
79
views
Probability of picking balls from 3 boxes without replacement [closed]
This problem has been come just from curiosity. Simple combinatorial arguments seem really complicated.
I have 3 boxes each having 75 green (G) balls, 20 red (R) balls & 5 blue (B) balls.
Each ...
- 1,305
2
votes
1
answer
82
views
Probability that one weighted mean of iid random variables is greater than the other
I read somewhere that if $X_1,\dots, X_n,Y_1,\dots,Y_m$ are all i.i.d. and admit probability densities w.r.t the Lebesgue measure and we choose weights $\omega_1,\dots,\omega_n,\rho_1,\dots,\rho_m$ ...
- 53
2
votes
1
answer
189
views
Uniform integrability of rescaled sample mean
Assume that $X_1, X_2, ...$ are independent and identically distributed random variables with mean $\mu$ and variance $1$, then let $\bar{X}_n=n^{-1}\sum_{i=1}^n X_i$ be the sample mean. We all know ...
- 1,766
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votes
1
answer
79
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Need help in understanding how to Calculate Estimation and Variance
I have this problem at hand and have no idea on how to approach it. Any leads/solution would be appreciated.
You roll a fair $6$-sided die, and flip a coin n times where n is the number on the die. If ...
2
votes
0
answers
361
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Implications of Girsanov Theorem
I am confused by the role Girsanov Theorem plays in deducing absolute continuity of laws of certain processes. Say $B$ is a Brownian motion in $(\Omega, \mathcal{F}_{\infty},\mathcal{F}_t,P)$ and let $...
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104
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Are there any results similar to the Paley–Zygmund inequality, but for probability density functions?
For a random variable $X$, the Paley-Zygmund inequality is given by
$$P(X>\theta \text{ E}[X])\geq\frac{(1-\theta)^2\text{ E}[X]^2}{\text{Var}{X}+\text{ E}[X]^2}.$$
This may be strengthened by the ...
- 175
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votes
0
answers
51
views
Winning probability for throwing one and two balls
Players $A$ throws one ball and player $B$ throws two. They are equally skilled and whichever ball gets closest to the hole wins. What is the probability that player $A$ wins?
My idea is to consider ...
- 600
2
votes
1
answer
71
views
Asymptotics of expected number of draws until repeat
Suppose there are $n$ distinct balls in a bag and they are drawn with replacement until the first repeat. Let $X$ be the number of balls drawn. I have shown that the distribution is unimodal and that ...
- 1,766
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0
answers
482
views
Lottery Probability Questions
In a game of chance, you mark two different numbers from 1 to 25 on a ticket. During the draw, two different numbers are drawn from an urn containing balls with the numbers 1 to 25. If you have marked ...
- 363
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0
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57
views
Probability never an equal number of black & white balls drawn is ${{m - n}\over{m + n}}$
Here's a question from my probability textbook:
A bag contains $m$ black balls and $n$ white balls ($m > n$). These are drawn out in succession. Show that the chance that there shall never be an ...
- 1,415
2
votes
0
answers
167
views
Compactness of the sublevel set of KL divergence---for the second argument, i.e., KL(P|| .)
Consider a completely metrizable sapce $\mathcal{X}$ and its borel sigma algebra $\mathcal{F}$. Let $PM$ be the collection of all probability measures defined on $(\mathcal{X}, \mathcal{F})$.
It is ...
- 21
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answers
55
views
A balanced coin is tossed three times. Define the events: Are events $A, B, C$ mutually independent?
A balanced coin is tossed three times. Define the events
$A=$the same result is obtained in the first toss and second toss.
$B=$the same result is obtained in the second toss and third toss.
$C=$the ...
- 850
2
votes
0
answers
102
views
maximum-likelihood function for the SST distribution
The SST distribution is a reparametrization of $ST3$, which is the skew t-student type 3. Below is some information.
Let $Z_0 \sim ST3 (0, 1, \nu, \tau)$ and $Y = \mu + \sigma\left(\frac{Z_0 - m}{s}\...
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0
answers
98
views
Ordered games problem
Suppose there are two bots (call them Bot 1 and Bot 2) playing a fighting game match. Each match consists of the bots playing a sequence of rounds, where each bot picks a particular character in the ...
- 115
2
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0
answers
212
views
Minimal Sufficient Statistics theorem 6.6.5 C&B proof
The theorem goes as follows:
Suppose the family of densities $\{f_0(x), .., f_k(0)\}$ all have common support. Then,
(a)
$$T(X) = \Big(\frac{f_1(X)}{f_0(X)}, \frac{f_2(X)}{f_0(X)}, .., \frac{f_k(X)}{...
- 1,313
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53
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For a stochastic process $X_n$, is $E[X_{n+1}]\leq X_n$ a stronger statement than $E[X_{n+1}|X_n] \leq X_n$?
I think that if $X_n$ is a supermartingale then it implies the unconditional version in the title, but not sure about the reverse, I suspect not.
- 6,908
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0
answers
119
views
Expected time to visit $a$ different nodes in a random walk on a directed graph
Given a $n$ vertices $m$ edges connected graph $G$ and a source vertex $x$, I start a random walk from $x$ until it discovered $a$ vertices for some small $a$ ($< n^{1/4}$). What is the expected ...
- 21
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0
answers
334
views
Probability Theory (Geometric)
A certain type of item produced by a factory has a 6% chance of being defective. Draw a random sample until you get the first defective. Let XX be the number of items that are drawn.
Find P(X=2).
I ...
- 65
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0
answers
53
views
Distribution of harmonic mean in a uniform triangle?
Suppose I have a triangle in the positive quadrant given by y=a-bx.
I have N points $(x_1,y_1),…,(x_n,y_n)$ uniformly distributed in this triangle. I’m interested in finding the point that minimizes ...
- 567
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0
answers
87
views
Would you ‘buy’ a prediction from Prof. Gott?
As a young man Mr Gott visits Berlin in 1969. He’s surprised that he cannot cross into
East Berlin since there is a wall separating the two halves of the city. He’s told that the wall was erected
8 ...
- 363
2
votes
0
answers
59
views
Can it ever be that for a random sample $X_1, ..., X_n$ we have that $\frac{1}{n}\Sigma_{i=1}^n x_i^2 \lt (\frac{1}{n}\Sigma_{i=1}^n x_i)^2$
I have had a homework problem about using method of moments for estimating a uniform random variable.
I probably made a calculation mistake because as was pointed out to me, we've shown in some ...
- 675
2
votes
0
answers
49
views
Expected semi-perimeter in Mondrian tessellation process
I'm working on Mondrian Process
[paper], which in few words splits a boxed region in $R^d$ by axis-aligned hyperplanes, uniformly located on a random axis, chosen proportionally to the lenght of the ...
- 31
2
votes
0
answers
52
views
Borel-Cantelli difficult summation
Let $X_1, X_2, \ldots$ be i.i.d. random variables with $P\{X_k=1\}=p$ and $P\{X_k=0\}=1-p$. Let
$$
R_n:=\sup\{k \geq 0 : X_n=X_{n+1}=\cdots=X_{n+k-1}=1\}.
$$
That is: $R_n$ is the length of the run of ...
- 627
2
votes
0
answers
135
views
Random number generator baseball
From the HMMT competition here: https://s3.amazonaws.com/hmmt-archive/november/2017/HMMTNovember2017ThemeRoundSolutions.pdf
New this year at HMNT: the exciting game of RNG baseball! In RNG baseball, ...
- 1,415
2
votes
0
answers
68
views
nonhomogeneous Poisson process elementary question
Let $N_t$ is a nonhomogeneous Poisson process. Find $P(N_{t_2}=n|N_{t_1}=m)$ where $t_1<t_2$ and $n \geq m$. My solution:
\begin{eqnarray}
P(N_{t_2}=n\mid N_{t_1}=m)&=&\frac{P(N_{t_2}=n,N_{...
2
votes
0
answers
51
views
Prune length distribution of random binary tree
Consider a random binary tree with $N$ leaves. Each node (except the root node) has a degree of exactly three (two children and one parent). No further restriction is placed on the structure of the ...
2
votes
0
answers
99
views
Regularly varying distribution function
Let
$$F(x)=2-2\Phi \left(\sqrt{\frac{1}{x}} \right)$$
with $x>0$ where $\Phi$ is the normal cummulative distribution function
a) Show that $G(x)=1-F(x)$ is regularly varying.
I'm having ...
- 137
2
votes
0
answers
71
views
How to get a law of large numbers?
I have a continuous-time stochastic process $X=\left(X_{t}\right)_{t \geq 0}$ for which I showed that there is $v>0$ such that, for integer times,
$$\frac{X_n}{n} \to v, \,\, \text{ $\mathbb{P}-a.s....
- 309
2
votes
0
answers
355
views
Why non-zero finite variance in chebyshev inequality.
I read Chebyshev's inequality from certain places. The statement is:
Let X be a random variable with a finite mean denoted as $\mu$ and a finite non-zero variance, which is denoted as $\sigma^2$, for ...
- 401