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].
100,487
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Convergence of the central limit theorem: pointwise vs uniform convergence
Let $X_1, X_2, \dots $ be a sequence of iid real random variables with finite moments (assume mean $\mu$ and variance $\sigma^2$) and call $X=\frac1N \sum_i X_i$.
The central limit theorem tells us ...
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8
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Density given two constraints
Suppose that $a,b $ are real constants. That $X_1, X_2$ are independent real random variables with densities $\phi_1, \phi_2$ with respect to the Lebesgue measure.
I know that a random variable $Y$ ...
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Quantitative analysis Statistical test of Likert Scale to determine best option from multiple options
What is the best statistical test to be done to determine the best drawing of three drawings?
Given data collected from a Likert scale on various characteristics of the drawings. Questions on ...
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1
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31
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Why do singleton events imply sets in multiplicative but not in additive probability?
Let $a \geq 0$ and $0\leq b \leq 1$ and $M, N$ be two appropriate conditioning events such that, for all singletons $y = \lbrace y \rbrace$ in the sample space $Z$ and all subsets $Y$ of $Z$, the ...
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A paradox-like consequence of Borel-Cantelli's Lemma
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables, whose range is $\mathbb{N}$ (e.g., Geo(1)). Consider now the random sequence $X_1,X_2,X_3,...$
What is the probability that each $...
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20
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Solving the probabilities of a combinatorial optimization
The optimization is to find $X$ to maximize
$$\max_X\|(I-\alpha X)^{-1}(X\circ A E)\|_F$$
where $X,A$ are $n$ by $n$ square matrices, $E$ is a vector $E=(1,\cdots,1)^\top$. More specifically, consider ...
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9
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How to determine number of entries in buckets of a perfect normal distribution?
Say you have a sample (n = 50,000) of student test percentages that range from 0% to 100%. If I create 5 buckets [0,20) [20, 40)..., how would I figure out the number of students to put in each bucket ...
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16
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Sufficient notion of stochastic dominance and conditional expectation
Consider two random variables $X$, $Y$. What are some notions of stochastic dominance for $X,Y$ that guarantee that $E[X \mid X> t] > E[Y \mid Y>t]$ for any $t$?
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How can I show that $(\|B_t\|^2-dt)_{t\geq 0}$ is a martingale?
Let $B$ be a $d$-dimensional $(\Bbb{F}, \Bbb{P})$-Brownian motion where $\Bbb{F}=(\mathcal{F}_t)_{t\geq 0}$. Then consider $X:=(X_t)_{t\geq 0}=(\|B_t\|^2-dt)_{t\geq 0}$. I want to check that it is a ...
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If sample average converges almost surely in an iid sample, must it converge to the mean?
SLLN tells us that if $X_1,...,X_n$ are iid, with $X_1$ having finite mean $\mu$, then their sample average converges a.s. to $\mu$.
However, suppose instead we know that $X_1,...,X_n$ are iid and ...
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If $\mathbb EX=0$ and $\mathbb E|X| < \infty $ implies finite Variance?
I think, I am totally wrong, but
if $\mathbb E|X|$ is finite then,
$$\text{Var}|X|=\mathbb E|X^{2}|- (\mathbb E(X))^{2}=\mathbb E|X^{2}|< \infty$$
Considering $\mathbb E|X|< \infty$ implies $\...
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Probability and Random Variables.
Hi, I was trying to understand this example in the book. In the first part of the question, We've to find p.d.f (probability density function). For that, we take the derivative of the given ...
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Conditional Probability from textbook example unclear
I am reading "Mathematical Statistics with Applications", 7th edition from Wackerly, Mendelhall and Scheaffer.
On example 2.23 on page 71, it is unclear how they calculate P(A|B).
Example 2....
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How to calculate Random Distribution?
I'm pretty sure this is a basic question but I can't seem to find an answer anywhere else.
If I have N objects each given one of M random categories (using a simple uniform random distribution), how ...
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Large deviation principle for small values of the maximum of the absolute value of a random walk
Let $X_i$ be an i.i.d sequence of symmetric real random variables with variance 1 and $S_n = X_1 + \dots + X_n$. Let $Z_n = \max_{1 \leq k \leq n } (|S_k|)$.
$Z_n$ is typically of the order of $C \...
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Binomial distribution probability (forming an equality)
The probability that Sally will play football in any week is 0.29. Sally
does not play football more than once in any week. The probability that
Sally play football at least once in a period of $n$ ...
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1
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95
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Find $P(X=k)$ where $X$ is the random variable that represents the number of draws required to obtain a white ball
Consider a box containing one black ball and one white ball. Every time a black ball is drawn, a die is rolled and a number of black balls equal to the result of the die are added to the box. We want ...
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Probability of extracting two numbers $a,b$ from $n$ numbers such that $|a − b|$ is $1$
This problem is from the first sheet of homework exercises from an Introductory Probability Course. We extract $2$ numbers $a,b$ from a box containing $n$ balls numbered from $1$ to $n$. We have to ...
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11
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Integral with respect to the Brownian sheet/Wiener Field [duplicate]
Let a Brownian sheet/wiener field, $W\left(t_1, \ldots, t_d\right)$, for some $d \in \mathbb{N}$ ( see this : https://encyclopediaofmath.org/wiki/Wiener_field for a definition).
For the $d=1$ case, we ...
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Limit of Pdf of an RV within it's support. [closed]
What is the proof that a pdf can never tend to positive infinity within it's support?
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49
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Probabililty of sums of gaussians
Let $X\sim N(\mu_1, \sigma_1)$ and $Y\sim N(\mu_2, \sigma_2)$. Consider an arbitrary $r\in\mathbb{R}$. How can I compute the probability of $\mathbb{P}(X \leq r \leq X + Y)$?
My idea: One could write ...
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Raffle - Odds of winning as one person/multiple tickets vs multiple persons/one ticket
Each week there is a lottery. X tickets are sold. There are Y people buying those tickets. There are N winners in one week. Buyers of the tickets are identified. If you buy Y tickets as a same person, ...
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10 consecutive flips of a coin by two players [closed]
A coin is flipped 10 times by player A and by player B separately . Player A wins only if the number of Heads he obtains is strictly bigger than the number of Heads obtained by player B. Probability ...
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Is posterior probability affected by a hidden observer?
I am not a major in mathematics. And I only have some basic understanding of probability. This is just a question that has bothered me for a long time in my understanding of posterior probability. So ...
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probability in higher-dimensional space
I have a question about probability.
In the "3-dimensional space + time in one direction" in which we live, the probability that a fair coin will turn heads is uniquely fixed at 1/2.
What ...
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Condition for $f^\prime$ to be absolute integrable
Suppose $f(x)$ is the probability density function of a random variable $X$, which means:
$$\int_{a}^{b} f(x) dx = 1$$
Also suppose $f$ is continuous and differentiable.
Provide a non-trivial ...
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The difference between the sum of the squares of the diagonal elements of WAW (for Wishart matrix W) and matrix A （for any A）
For any symmetric matrix ${ A}\in R^{K\times K}$, we can compute ${ B} = { W}{ A}{ W}$ where $W\in R^{K\times K}$ is a Wishart matrix with $N$ degrees of freedom (N>K).
I want to bound term $\sum_{...
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Mathematics Behind Coincidence of Probabilities (Magic the Gathering Example)
In Magic the Gathering, a player must decide how many land cards to include in his or her deck. Drawing too few at the beginning of a game is called mana screw. Drawing too many at the beginning of ...
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Probabilistic interpretation of Fourier Transform
The fourier transform is given as below :
$$F(\omega) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty} f(t) e^{-i \omega t} \mathrm{d}t$$
Now $$F(\omega) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty} ...
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Expected value of a multivariate normal distribution given a linear constraint
I saw this problem: you toss 10000 coins, 5000 silvers, 5000 golds. If you get head up on a gold coin. you get 4 dollars. If you get head up on a silver coin, you get 1 dollar. Given you get 14000 ...
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Breaking a stick into three pieces - expected length of longest piece
Same question as here but I'm trying to work out what's wrong with my logic. The question is:
Take a stick and break it randomly into three pieces (i.e., two
randomly placed breaks on the stick). ...
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Perplexing Probability Combination Question [closed]
A bag contains 4 red and 5 black identical balls.
5 balls are selected at random one after the other without replacement find the probability that a red ball was picked three times
I've used ...
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When does the product of random matrices diverge?
Suppose $A_i$ are IID samples of a random matrix-valued variable. I'm interested in determining whether the following infinite product is likely to diverge
$$A_1 A_2 A_3\cdots$$
Finding necessary+...
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Generating Function for Coin Flip followed by a roll of two dice
I'm self studying probability, specifically generating functions. I'm searching for the relation between conditional probabilities and probably generating functions in order to justify writing the ...
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Conditional Probability of Biased Coin Toss
I am currently self-teaching myself probability because through MIT opencourse ware because my teacher merely reiterates the textbook in an even more complicated way.
In Lecture 3.6, the instructor ...
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Drawing balls until most of the colours have been drawn
You have an urn with $N$ balls of different colours. There are $M$ colours and $N/M$ balls of each colour. Draw balls from the urn without replacement. Let $X_n$ be the number of different colours ...
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Markov Chains $r_{ij}$ and $E[N_j|X_0=i]$
I am given a transition matrix equal to
$$P = \begin{bmatrix}
0 & 0 & 0 & 3/4 & 0 & 1/4 \\
1/2 & 0 & 1/2 & 0 & 0 & 0 \\
0 & 2/3 & 0 & 0 & 0 &...
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Prove that at least $4$ persons are common
In an assembly of $30$ persons, $300$ groups are formed. Each group contains $10$ persons each. Prove that one can find two groups having at least $4$ common members.
My approach is that first I'll ...
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If a sequence of random variables converges in probability to $0$, will the sequence still converge to $0$ when multiplied by $\sqrt n$
Assume we have a sequence of random variables $X_1,X_2...$ such that the sequence converges to $0$ in probability. Under what general conditions will the sequence still converge in probability to $0$ ...
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Conditional Expectation as the best approximation
I saw this question in a book at the university and did not manage to solve it.
let X,Y be discrete r.v with finite expectation and variance. Show
that for every h:R->R
E[(X-E[X|Y])^2 ] <= E[(X-...
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Truncating the pdf function
Sorry for the naive question, but I can't figure out the answer to.
Edited for clarity
I have a dataset 𝐗 and I'm sampling from this dataset, evaluating the samples on my normal distribution $P$ pdf ...
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Probability Distribution for the Difference of Random Variables?
I was reading this link on Probability Distributions of Random Variables (https://www.probabilitycourse.com/chapter6/6_1_2_sums_random_variables.php) and had the following question on the distribution ...
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Conditional probability in multinomials? [closed]
Assume next quarter's offering of a class called STATS 101 is exactly 600 people, that each of the four undergraduate classes is comprised of 1750 students, and that next quarter's STATS 101 roster is ...
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Show that it is a stopping time
My task is to describe the following stopping time formally and to prove that it is indeed a stopping time.
Let $A\in \mathcal{B}(\mathbb{R})$ be a fixed set. $(X_t)_{t\ge 0}$ hits A for the first ...
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Ito diffusion crosses initial value infinitely many times?
Consider the stochastic process $Y_t=y+\int_0^t\sigma(Y_s)dW_s$ where $W$ is a standard Brownian motion. It is well known that if $\sigma(y)=\sigma >0$ constant we have that $Y_t$ crosses $y$ ...
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The estimator of the capability indices
The capability indices $C_{pk}$ and $P_{pk}$ are defined for a normally distributed random variable $X$ with mean $\mu$ and standard deviation $\sigma$ and specification limits $-\infty <LSL < ...
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A recursive sequence which converges in probability but its limit doesn't exist [closed]
Suppose $\left(Y_n\right)$ is a sequence of i.i.d. random variables such that
$$
\mathbb{P}\left(Y_n=1\right)=\mathbb{P}\left(Y_n=-1\right)=1 / 2 .
$$
Let $\mathcal{F}_n=\sigma\left(Y_1, \ldots, Y_n\...
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16
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Gaussian concentration to upper bound the variance of the norm of a gaussian vector
I want to prove the following:
Let $X_1$ be a centered Gaussian vector in $\mathbb{R}^d$ with covariance matrix $\Sigma$.
Then:
$ Var( \|X_1 \|) \lesssim \left(\mathbb{E} \|X_1 \| \right)^2 $
This is ...
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Using Mass distribution principle to provide lower bound for the Hausdorff dimension of Cantor set
given the (ternary) Cantor set $\mathcal{C}$ it is well known that its Hausdorff dimension is given by $\dim_\mathcal{H}(\mathcal{C})=ln(2)/ln(3)$, which I am going to denote by $\alpha$. I am ...
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Calculating Prize Line Expectation Part 2
Thanks in advance for any help.
Yesterday a very helpful member called @joriki answered my original question on this and that conversation came to a conclusion as a result. I have a second part that ...
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