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].
100,508
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Expected time to get on bus
Suppose buses arrive at the bus stop according Poisson process with rate $\lambda$. You get on bus if it is not full. The probability that a bus is not full is $p$ and is independent of arrival time. ...
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Can the Law of Large Numbers be appied in Number Theory?
Usually, statistics have no place in Number Theory, but the Law of Large Numbers can be an exception, since it strictly deals with infinite cases.
For example, if one throws a dice a finite number of ...
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Rejection Sampling in Poincaré disk - mapping a pdf to the tangent space
I want to map the pdf of the Normal distribution given in Pennec to its tangent space, given mean $\overline x$ and precision matrix $\Gamma$. The Normal law on a manifold $\mathcal M$ is given as $$...
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sum, max and difference of dependent discrete random variables
I am trying to calculate the following random variable distribution:
$Z= \sum _{i=0} ^{n} [max(X_0, X_1, X_2,...,X_n)-X_i]$
I know $X_i$ probability distribution ($X_i$ are i.i.d. random variables), I ...
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Finding the median of the total number of die rolls
Question
A fair die is to be rolled repeatedly until a six comes up. Find the median of the total number of rolls given that five comes up on the first roll.
My working
Clearly, this follows a ...
- 2,095
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25
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Playing a fair dice and find expectation
Alice and Bob play the following game with a fair dice.
Alice rolls the dice six times, and wins
Rs. 100 if she gets a six in at least one of the six rolls; she wins nothing in all other cases.
Bob
...
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22
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How to combine antithetic variable and control variate methods in mote carlo integration?
I want to use R to estimate the integral $\theta=\int_{0}^{1} e^{x^2 }\,dx$ by monte carlo integration with variance reduction. The variance reduction I want to use is combining the antithetic ...
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How do I integrate the likelihood of successfully hitting a target as it closes in from a distance with a known jitter (inherent imprecision)?
Very rough C# estimation
https://pastebin.com/tq3d2TEr
It appears that assuming a perfect lock on a non-dodging missile that shootdowns are rather likely. I'll add more to this soon.
Premise
I'm ...
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Apply Equal (Unknown) Transformation to Two Random Variables
I am looking for an algorithm (if one even exists) that will accomplish the following, or something similar:
PARAMETERS:
Unknown cdfs $F$, $G$
Known target cdf $F'$
Unknown, increasing bijection $\...
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1
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66
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Definition of probability density
I know that every probability density function (pdf) must be nonnegative and integrate one on $\mathbb R$.
But the definition of pdf requires a distribution function $F$: "we say that a function $...
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Finding CDF and PDF of random variable Z=X/Y
$$F_Z(z) = \int_0^{\infty} \int_0^{yz} f_X(x) f_Y(y) dx dy$$
for Y [Ymin,Ymax] and X>0
$$F_X(yz)F_Y(y) \Big|_Y^{Ymax} - \int_Y^{Ymax} F_Y(y) dF_X(yz) $$
Can I write an equation ...
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43
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Probability Each Person Gets An Ace [duplicate]
This probably sounds like a really stupid obvious question, but bear with me. I have attached the problem from a lecture from MIT Intro to Probability course. I simply do not understand why ...
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31
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uneven probability - how to detect rigging?
Imagine a video game in which the players has some percentage chance to hit a shot. (XCOM, for example)
The percentage chance presented for each shot is dependent on various environmental factors such ...
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23
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How to calculate the probability of a repeated result, given x chance that an event occurred in the first place?
To calculate the probability of a random roll repeating after N rolls is derived from the Birthday Problem, but how does that calculation change if there's only a certain chance of having a birthday ...
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35
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Calculating Probability of a horse winning based on expected position
My question is this:
If you have 4 horses (A , B , C and D ) in a race:
Horse A is expected to get position 2
Horse B is expected to get position 2
Horse C is expected to get position 3
Horse D is ...
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How do I calculate the following probability?
Let's say there are 5 cats and 20 dogs and you get a random group of 2 animals. What is the probability that you end up with a dog and a cat? All possible combinations where you get a cat and a dog ...
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26
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Probability of an event occurring at one time of day vs another
I have a dataset of events that occur throughout the year at different times of the day. The events are tallied in the morning and afternoon. I would like to test the probability of the event ...
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How can I determine the needed sample size for populations with a large amount of possible values?
Say there is a set of numbers of size $N$ where $N > 1,000,000,000$, and for each $i$ in $N$: $0\leq i\leq 1000$. In other words, there are over a billion numbers between $1$ and $1,000$.
I need to ...
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Joint distribution given marginals and correlation
I am trying to understand whether it is possible and if so then how to find a joint distribution given marginals and a correlation matrix.
In particular, suppose $X_{1},\ldots,X_{n}$ are discrete ...
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Laplacian Rule of Succession. How to find $P(X_{n+1}=1 \mid S_n=n)$?
Been working through stat110 and "https://youtu.be/N8O6zd6vTZ8?t=2990" talks about the laplcian rule of succession and I am not too sure how he got to his final step(49:50). The problem is ...
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bivariate normal distribution mean, standard deviations
Let (X,Y) have the joint bivariate normal distribution with probability density function as
f(x,y)=(1/(15Π))exp{-2[((x-2)²/9))+((y-4)²/25)-((x-2)(y-4)/9]}; -∞<x<∞, -∞<y<∞
What will be ...
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If 𝑋 and 𝑌 are two independent random variables and 𝑋 and g(X)+𝑌 are independent then g(X) must be a constant almost surely.
Let $X$ and $Y$ be two independent random variables and $g$ be a measurable function. Assume that $X \, \bot \, g(X)+Y$. Does this imply that g(X) is almost surely constant?
My intuition is that this ...
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32
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What is the probability of picking a point in the edge of a ball?
How can we describe the probability law of a point in the edge of a ball?
First I thought about something like:
If $ r $ is the radius of the ball, then maybe I can try the volume formula:
$$ \int_0^r ...
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24
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Looking for help with an actual solution to a stepping-stone probability problem
Back in college, I remember a professor giving us this question:
In front of you is an endless line of stepping stones. You flip a coin: on heads, you step forward one stone (onto Stone 1), and on ...
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19
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Distribution of a $M$-dimensional random variable that is binomially sampled from another $N$-dimensional random variable, when $M \leqslant N$
Given a continuous N-dimensional random variable $X\sim P(\textbf{x}), X\in R^N$, $\textbf{x}=(x_1,x_2,x_3,...,x_N)$. For example, $X$ is a $H\times W$ Image, where $H\times W = N$. Now given the ...
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Triviality of the Chernoff bound for $t$ less than the expectation of the random variable
Let $X$ be a real valued random variable and $\lambda \ge 0$ . Then for any $t$, we have $P(X\ge t)\le e^{-\lambda t}\mathbb{E[e^{\lambda X}}]]$.
Now we can get the best possible bound of this type ...
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19
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Uniform distribution of product of prime numbers modulo
Let $\{p_i\}$ denote the sequence of all prime numbers, minus a given prime $q$. Let $\{B_i\}$ be i.i.d. Bernoulli random variables. Consider then the random variable $X_n = \prod_{i=1}^n p_i^{B_i}$, ...
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57
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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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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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41
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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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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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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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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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1
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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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44
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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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22
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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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27
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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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112
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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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40
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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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24
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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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1
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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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32
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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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1
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38
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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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