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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1answer
23 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
23 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
14 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
24 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
0answers
18 views
Is there an expression for a CDF of one random variable with respect to another random variable, given a condition? [duplicate]
Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as:
$$\mathbb{P}[X < Y] = ...
0
votes
2answers
46 views
Sample $x$ from $g(x)$
I got confused with all this randomness and probability functions. I was trying to implement the rejection sampling method which (apparently) is really simple. I was reading from Rejection Sampling in ...
-1
votes
1answer
67 views
Finding an expression for the probability that one random variable is less than another, given a condition.
Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as:
$$\mathbb{P}[X < Y] = ...
0
votes
1answer
31 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 ...
1
vote
2answers
73 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 ...
0
votes
0answers
23 views
Replicating portfolio under the Black-Scholes model
I have a two-asset Black-Scholes model:
$dB_t = B_t r dt$
$dS_t = S_t (\mu dt + \sigma dW_t)$
I introduce a European claim $\xi = \max(K,S_T)$ with maturity $T$, for some fixed $K$. I have ...
4
votes
1answer
46 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
35 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
60 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
24 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?
3
votes
1answer
43 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 ...
3
votes
1answer
20 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
67 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
1answer
56 views
Probability of A occurring before B
If event $A$ occurs with probability $p_A$ in a time period of length $t$, and $B$ occurs with probability $p_B$ in a time period of length $t$, what is the probability that $A$ will occur before $B$?
...
0
votes
0answers
17 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 ...
2
votes
0answers
25 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
25 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
1answer
94 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 ...
0
votes
1answer
18 views
probability specific people appear in a sample from a group
The Justice League wants to randomly select a group of 7 from among the 40 currently available superheroes/superheroines to investigate a glowing meteorite. What is the probability that Bat Man, ...
1
vote
1answer
30 views
Compute the mean of a random variable
Imagine I have for a single individual some variable $X$ with mean $\lambda$ (for example the number of cars he has). Now I take a whole population of individuals. The parameter $\lambda$ for each of ...
1
vote
0answers
36 views
A special random subset of uniformly distributed numbers is still uniformly distributed?
Assume that I have a value range [1,1000].
Goal: I want to have 10 numbers randomly sampled from [1,1000].
case1:
...
1
vote
1answer
29 views
Showing it is a joint probability density function
I have two random variables $X,Y$ with a joint density function $f_{X,Y}(x,y)=x+y$ if $(x,y)\in[0,1]\times [0,1]$ and otherwise $f_{X,Y}(x,y)=0$
I want to analyze this case in different cases, first ...
1
vote
2answers
57 views
Expected number of pieces of a chessboard
If n squares are randomly removed from a $p \ \cdot \ q$ chessboard, what will be the expected number of pieces the chessboard is divided into?
Can anybody please provide how can I approach the ...
1
vote
0answers
22 views
Bernstein type inequalities. Is there a standard list?
Suppose I have a sequence of iid random variables $X_i\geq 0$ with mean $\mu$ and $\mathbb E \left(e^{tX_i}\right) = G(t)$. Set $$S_n = \sum_{i=1} X_n.$$
For the purpose of this question the ...
1
vote
1answer
45 views
distribution function of time T
an ambulance station is located 30 miles from one end of a 100-mile road. the station services accidents along the entire road. suppose that an accident occurs. suppose that Suppose accidents occur ...
0
votes
1answer
13 views
Third central moment Bernoulli variable
I'm looking for a proof of the third central moment of a Bernoulli variable $X$ with probability $p$. I know it must be $p(1-p)(1-2p)$, but I'm looking for a way to show this. Any ideas? Thanks!
1
vote
1answer
39 views
Multivariate normal distribution density function
I was just reading the wikipedia article about Multivariate normal distribution: http://en.wikipedia.org/wiki/Multivariate_normal_distribution
I use a little bit different notation. If $X_1,...,X_n$ ...
1
vote
1answer
28 views
Generating points in rectangle
I am trying to generate $N$ points randomly and uniformly distributed in an $m \times n$ rectangle. How can this be done? I have tried to split the initial rectangle into as many rectangles i could, ...
0
votes
1answer
14 views
partials of a PDF with no closed form solution
I need to estimate partial derivatives for all N parameters denoted $\theta_{N}$ of a probability density function(PDF) $\mathcal{f}$.
This PDF $\mathcal{f}$ has no closed form solution and is ...
2
votes
1answer
42 views
Given that the family has at least one girl, determine the probability that the family has at least one boy.
Suppose that a family has exactly n children (n ≥ 2). Assume that the probability that any child will be a girl is 1/2 and that all births are independent. Given that the family has at least one girl, ...
1
vote
1answer
29 views
A problem on almost sure convergence
Consider a sequence of random variables defined on the standard unit interval probability space :
$ X_n = 2^n \text{when} \frac{1}{2^n} \leq \omega \leq \frac{1}{2^{n-1}}$
...
0
votes
1answer
49 views
Markov Chain - Snakes and Ladders
A simple game of snakes and ladders is played on a board of nine squares. At each turn a player tosses a fair coin and advances one or two places according to whether the coin lands heads or tails. If ...
0
votes
1answer
32 views
How to calculate the pmf of $X_N$
How do I calculate the pmf of $X_N$, where $X$ is the number of people out of $N$ getting back their own hat after a random hat exchange?
How can I calculate it without listing all the possible ...
0
votes
0answers
27 views
distribution function and density function
A lion is standing $30$ meters from one end of a $100$-meter road. The lion will attack any zebra that appears on the road. Suppose that a zebra appears on the road, and suppose that the position at ...
2
votes
1answer
20 views
If $x^p P(|X|>x|)=o(1)$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon<p$
If $p>0$ and $x^p P(|X|>x|)=o(1)$ as $x\to\infty$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon<p$.
It feels like the assumptions should lead to something like $\sum_n^\infty ...
0
votes
0answers
29 views
Number Plate Problem
I'm having trouble with a question that seems to perplex:
A number plate contains three letters followed by three numbers. A number plate is selected at random.
Calculate the probability that the ...
2
votes
2answers
52 views
$ E\left( \left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \right) \le \left( \frac{1}{n}\sum_{j=1}^n E(|X_j|^p)^{1/p} \right)^p$
The following is problem 14 of section 3.2 from Chung's "A Course in Probability Theory".
If $p>1$, we have
$$\left| \frac{1}{n}\sum_{j=1}^n X_j \right|^{p} \le \frac{1}{n}\sum_{j=1}^n |X_j|^p$$
...
1
vote
1answer
18 views
Marble Possibility P(At least one yellow)
There are $2$ black and $3$ yellow marbles in a bag. $2$ marbles are drawn randomly without replacement. What is the possibility that at least $1$ yellow marble is selected.
-1
votes
1answer
27 views
How to compute conditional expectation of a log function
I've been studying the Expectation Maximization algorithm. According to the formula shown here, what I have to do in the M step is to compute a new $\theta$ that maximizes the conditional expectation ...
-3
votes
1answer
47 views
Probability of purple party voters in a samaple prediction of two cities A and B
City A has 1,000,000 people; City B has 4,000,000 people. Suppose the goal is to try to predict the percent of Purple Party voters in a sample. Other things being equal, a simple random sample of 1% ...
1
vote
2answers
20 views
Probability of senior citizens in a one million residence
In a city of over $1000000$ residents, $14\%$ of the residents are senior citizens. In a simple random sample of $1200$ residents, there is about a $95\%$ chance that the percent of senior citizens is ...
1
vote
2answers
48 views
Monte Hall Problem - Question on Decision Tree Construction (Conditional Probability)
(The Monte Hall Problem, also known as the 3 door problem): On the game show Let's Make a Deal, there are 3 doors hiding 3 prizes. Two of the prizes are goats, 1 prize is an expensive new ...
1
vote
0answers
60 views
How to calculate probability with sigmoid output in feedforward neural network?
first of all I'm sorry for my not very skilled English, but I will do my best to explain my problem.
I'm trying to create a feedforward neural network with one hidden layer (with probably arctan ...
1
vote
1answer
31 views
Dice probability when a number is disallowed in the first round
In the game Settlers of Catan a player starts each turn rolling 2 six sided dice. There's a variation of the game where if a 7 is rolled in the first round of a game (a 'round' is when each player has ...
1
vote
1answer
23 views
Dice tracker and roll probability percentage
I have a dice tracker that graphs the combined roll of 2 fair six sided dice (numbers 2 through 12). I want to display the variance percentage that the current number is over/under as a ...
1
vote
0answers
29 views
Hypothesis testing from negative binomial data
For each trial I flip a coin until $3$ heads or $3$ tails occur, whichever comes first. Out of $10$ trials, $3$ trials result in $3$ flips, $3$ trials result in 4 flips, and $4$ trials result in $5$ ...
