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 ...

learn more… | top users | synonyms (2)

1
vote
1answer
77 views

The man with two boys [duplicate]

I have recently seen a probability question which says "i am asking randomly the persons I met if they are having two chidren and one of them is a boy who was born on tuesday. At last I met one whose ...
1
vote
1answer
97 views

Calculating the probabilities of different lengths of repetitions of numbers of length 4

I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
0
votes
1answer
25 views

Independence of transformed variables

There are two independent variables X and Y. Y is an input for non deterministic algorithm f, and the output of f(Y) is Z. How to prove that X and Z are independent?
0
votes
1answer
67 views

Expectation of function of stochast

I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
0
votes
1answer
480 views

Conditional Probability: Bayesian Cause/Effect Question

The probability that a randomly chosen male has a circulation problem is 0.25. Males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation ...
2
votes
1answer
95 views

Urns version of Laplace's law of succession

I'm trying to prove urns version of Laplace's law of succession my professor suggested. Laplace's law states that the chance that the next trial is a success given $j$ successes out of the first $n$ ...
18
votes
2answers
355 views

Generalized nontransitive dice

Let $X_1, \ldots, X_n$ be a collection of random variables. Consider the directed graph with vertex set $\{ 1, 2, \ldots, n \}$ where there is a directed edge $i \to j$ if $\mathbb{P}(X_i > X_j) ...
0
votes
1answer
182 views

Probability of two random n-digit numbers dividing each other

Let $n$ be a positive integer. Suppose $a$ and $b$ are randomly (and independently) chosen two $n$-digit positive integers which consist of digits 1, 2, 3, ..., 9. (So in particular neither $a$ nor ...
1
vote
0answers
85 views

Fast way to estimate cardinal number of subset

I have a large set $S$ of items, but the set is not exactly known. All I know are the cardinal numbers of categories i.e. a number of disjoint subsets, $ \vert{S_1}\vert \dots \vert S_n\vert$ with ...
0
votes
0answers
67 views

Estimating the radius of a circle

I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
1
vote
1answer
65 views

What kind of functions can be moment-generating functions for a random variable?

Given an infinitely differentiable function $ g: \mathbb{R} \rightarrow \mathbb{R}$, can we always find a distribution function $f_X$ of some random variable $X$ so that $g(t) = \int_{-\infty}^\infty ...
3
votes
1answer
466 views

Neyman-Pearson lemma on Normal distribution

We've got a random sample of iid $X_1,\dots,X_n$. We're testing the mean of $X \sim \mathcal{N}(\mu,\sigma^2)$, where $\sigma^2$ is known. The size of the test $\alpha=0.05$. $H_0: \mu=0$ $H_1: ...
1
vote
0answers
72 views

How does this violate probability theory?

Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$) $p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$ $p(Y = -1) = .5$, $p(Y = 3) = .5$ Question: Despite not being handed any information ...
1
vote
1answer
38 views

Convolution of r.v.'s

Suppose a sequence $\{X_{n} \}$ of pairwise independent r.v.'s. If $F_{X_n}$ is the distribution function of $X_n$ then $ F_{X_1 + \ldots + X_{N}} = F_{X_1} \ast F_{X_2 + \ldots + X_{N}}$ ...
1
vote
1answer
92 views

Conditional joint probability and independence

Let's have a joint probability of three events, $\mathbf{P}(X,A,B)$. If $\mathbf{P}(X|A) = \mathbf{P}(X)$, can we show that $\mathbf{P}(X|A,B) = \mathbf{P}(X|B)$? If so, how?
0
votes
1answer
74 views

$P\{X_t=-X_t \}=1$

If we define that $X_t$ is Brownian motion over space $(\Omega,\mathcal F ,\mathcal F_t;P) $, then why is it true that the fact that $X_t$ is Brownian motion implies that $P\{X_t=-X_t \}=1$ is ...
1
vote
2answers
210 views

Uniform distribution on the n-sphere.

I have the next RV: $$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$ where $$X_i \tilde \ N(0,1)$$ It's a random vector, and I want to show that it has a uniform ...
1
vote
1answer
2k views

Joint distribution of multiple binomial distributions

In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them. The original file can be ...
0
votes
1answer
35 views

Probability of all elements of a subset being coprime

Let $S=\{1,..,n\}$ and $R \subset S$ ($|R|=k$, $k<n$) -- $R$ is a random subset of $S$. Let $m=min(R)$, and $R'=\{x-m: x \in R, x \neq m\}$, so $|R'|=k-1$. What's the probability that ...
3
votes
3answers
315 views

Compute value of $\pi$ up to 8 digits

I am quite lost on how approximate the value of $\pi$ up to 8 digits with a confidence of 99% using Monte Carlo. I think this requires a large number of trials but how can I know how many trials? I ...
0
votes
1answer
127 views

Proof of Bienayme Inequality

I have a bit of trouble about the proof of Bienayme Inequality. Bienayme Inequality is as follows: If X has mean $\mu$ and variance $\sigma^2$, then $$\mathbb{P}\left(\frac{|X-\mu|}{\sigma}\ge ...
2
votes
0answers
63 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
2
votes
0answers
102 views

Probability distribution for a digit of a number

If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
0
votes
1answer
104 views

Continuous Non negative martingale converging to 0

Is there any (non trivial) continuous non negative martingale which converges to 0?
1
vote
1answer
47 views

Probability of catching subway.

A blue tram shows up randomly in a uniform distribution given any hour of the day at a certain stop. A person shows up independently within this same hour. If they are only willing to wait 10 minutes ...
2
votes
1answer
63 views

Finding a PDF from a function

I have a function $y = f(x),\ x\in\mathbb{R}$ (assume $f(x)= \sin(x)/x$ if you need an example). How can I find the probability distribution function (PDF) of $y$, assuming $x\sim U(\mathbb{R})$ ...
7
votes
2answers
497 views

Find: The expected number of urns that are empty

A total of $n$ balls, numbered $1$ through $n$, are put into $n$ urns, also numbered $1$ through $n$ in such a way that ball $i$ is equally likely to go into any of the urns $1, 2, . . . , i$. Find ...
0
votes
0answers
97 views

Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin

I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
0
votes
1answer
99 views

Approximation of a random variable by a sequence of simple random variables

It said in a probability book that any non-negative random variable $X$ can be approximated by a sequence of simple random variables (finite range) $X_1,X_2,\dots,X_n$ such that ...
1
vote
3answers
191 views

A simple probability reasing to predict rain fall

A friend told me the following about whether it will rain tomorrow (or not): The probability that it will rain tomorrow is $1/2$ since it will either happen or not. But -even as a non mathematician- ...
2
votes
1answer
85 views

Conditional Probability, Lack of Dependence on a Parameter

I am trying to understand why the following is true: $$ p(f(Y) = f(y) \mid Y = y) = p(f(Y) = f(y) \mid X = x, Y = y) \qquad \ldots \text{(Eq. 1)} $$ where $Y$ and $X$ are random variables, and $f(Y)$ ...
1
vote
2answers
138 views

Generalization of Doob Dynkin for Stochastic processes

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
0
votes
1answer
86 views

Finding an expression for a multi variate joint CDF.

Let $X,Y$ and $Z$ be random variables with $X$ and $Y$ dependent, and $Z$ independent of both $X$ and $Y$. Let $f_{X},f_{Y},f_{Z}$ denote the density function's of $X,Y$ and $Z$ respectively and ...
0
votes
1answer
38 views

Splitting multivariate normal into individual (correlated) components

I have a multivariate normal variable $X$ with mean zero and variance $\Sigma$. I would like to write every component $X_i$ of $X$ as: $$ X_i = \phi_i Z_i $$ where $\phi_i$ is a scalar and $Z_i$ is a ...
0
votes
0answers
35 views

A special random subset of uniformly distributed numbers is NOT uniformly distributed?

I asked the same question in the post: A special random subset of uniformly distributed numbers is still uniformly distributed? Let me describe my question again. Assume that I have a value range ...
0
votes
2answers
58 views

Sample $x$ from $g(x)$

I got confused with all this randomness and probability functions. I was trying to implement the rejection sampling method which (apparently) is really simple. I was reading from Rejection Sampling in ...
-1
votes
1answer
3k views

Finding an expression for the probability that one random variable is less than another, given a condition.

Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as: $$\mathbb{P}[X < Y] = ...
0
votes
2answers
117 views

Post-Uni Calculus/Probabilities Book Suggestion

I have a Computer Science Background, recently graduated and I would like to refresh/improve my knowledge about probabilities and statistics (also calculus). The priority is probabilities and ...
0
votes
4answers
294 views

Independent and uniformly distributed on $(\frac{1}{2},1]$

I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to ...
4
votes
3answers
951 views

Probability of at least N events occuring

I have a series of N events, each with its own probability of occurring. How would I calculate the probability that at least M of the N events actually do occur? I think this is conditional, in that ...
0
votes
1answer
742 views

Accept reject method to generate random numbers

The method says that having a proposal $g(x)$ Sample $X^* \tilde ~ g(x)$ and $U \tilde ~ Unif(0,1)$ Accept $X = X^*$ if $U ≤ f(X^*) / M g(X^*)$ Moreover, $M$ is constant that satisfies $Mg(x) ≥ ...
6
votes
2answers
112 views

$\lim_n \frac{1}{n} E(\max_{1\le j\le n} |X_j|) = 0$

If $\{X_n\}$ is a sequence of identically distributed r.v.'s with finite mean, then $$\lim_n \frac{1}{n} E(\max_{1\le j\le n} |X_j|) = 0$$ The inequality $$\frac{1}{n}E(\max_{1\le j\le n} |X_j|) ...
0
votes
1answer
202 views

Combination of arrangement and probability

Four guys and four girls are arranged in a row such that no two girls are together. What is the probability that any two of the four guys are together?
5
votes
1answer
115 views

hint with Bayes rule problem

The pirate Captain Queequeg has a lazy crew and suspects they are planning to stage a mutiny. Captain Queequeg's solution is to have every member of the crew roll Queequeg's lucky die. If the roll is ...
4
votes
1answer
582 views

Bound on expectation of absolute value in terms of variance

In my book it says that a white noise process $\{Z_t\}$ with mean zero and variance $\sigma^2$ has the following property: E$|Z_t| \leq \sigma$. This had me thinking of Jensen's inequality, that ...
4
votes
3answers
132 views

A basic doubt on Lebesgue integration

Can anyone tell me at a high level (I am not aware of measure theory much) about Lebesgue integration and why measure is needed in case of Lebesgue integration? How the measure is used to calculate ...
0
votes
0answers
33 views

Marginal Pdfs for Continuous Random Variables

http://oi42.tinypic.com/ddyjph.jpg this problem is confusing me, i know how to start it, we need to find $f_Y(y)$ so we integrate with respect to x and i get $-2e^{-x}e^{-y}|^y_0$ which then should ...
6
votes
1answer
183 views

lower bound of expectation of stochastic differential equation

I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
0
votes
2answers
40 views

Central limit theorem - std dev away from mean

I was reading about the CLT and found something that I think people use interchangeably. On one hand I found that 68% of the means are 1 standard deviations from away and 95% are 2 std dev. On the ...
10
votes
2answers
246 views

Edge percolation on $\mathbb{Z}^2$: probability that two neighbouring vertices are connected?

I'm considering edge percolation on $\mathbb{Z}^2$ with parameter $p$, so that edges are present with probability $p$. Is it known how to express the probability $P(p)$ that $(0,0)$ is in the same ...