Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.
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9 views
This is regarding hypothesis testing using F-distribution and Chi square distribution
Two ambulance stations, A & B, are in similar locations. Random samples of the response
times, in minutes, to emergency calls were recorded during a particular week. The
information is given in ...
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votes
1answer
24 views
This is regarding inomial distribution
Two acceptance sampling schemes, A and B, are proposed for deciding whether or not
to accept a large batch of items from a production process in which 5% of the items
produced are defective.
Scheme A: ...
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1answer
15 views
Basic question on the transformation of Exponential distribution.
Why central moments coincide for random variables
$V\sim E(a,h)$ and $Y\sim E(h)$
where a=location parameter
h= scale parameter.
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0answers
24 views
Convolution of exponential distribution.
let $Y_1\sim E(λ_1 )$ and $Y_2\sim E(λ_2 )$. $Y_1$ & $Y_2$ are independent random variables
let $V=Y_1+Y_2$
show that the pdf of $p_V(x)$ = $\frac {\exp[-(x/λ_1)] - \exp[-(x/λ_2)] } ...
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0answers
11 views
Problem with normal distribution and change of variable.
I´m finding this problem really hard to solve, i hope someone could help me. The problem is:
Given X, random variable, with normal distribution N(0,1) (standard). And given that Y=X^2
Find the ...
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0answers
21 views
Proof that the radius is a sufficient statistic for a circle
How can I prove that the radius of a circle is the sufficient statistic for the probability of choosing random points in the area of the cricle?
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0answers
6 views
probability to cover straight line with circles
suppose sensors of homogeneous radius r are dropped by Poisson distribution on a straight line of length L. how to calculate that the straight line is covered by sensors with probability P
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0answers
20 views
Stochastik and Bernoulli experiment
My Task is:
You got N coins, where N is distributet by a Poisson-distribution with the Parameter lambda.
Now we got the likelihood for head of 0 < p < 1. Where K are the heads that you got ...
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0answers
23 views
Weibull distribution - is it “heavy tailed” with three parameters? (prooving power law) [closed]
I am fitting my data to Weibull distribution (I would like to prove, that they follow power law).
If I use Weibull with two parametrs to my data, I get "shape parameter" greater than 1 - so no heavy ...
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votes
2answers
18 views
Probability of two variables of having the same value
Let $X$ and $Y$ be two random variables, whose PDFs $f_X$ and $f_Y$ are uniform. $f_X$ and $f_Y$ may overlap. For instance, they could represent two score distributions for two tuples $x$ and $y$ in a ...
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2answers
46 views
Probability Joint PDF
Every night Joe goes to the casino and takes with him an amount of money in dollars, X, that is distributed according to the pdf:
f(x) = Ax^2 for 0 < x < 10
where A is a constant that you need ...
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0answers
26 views
Calculating the probabilities of different lengths of repetitions of numbers of length 6
This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4
except now I'm having problem with numbers of length 6.
...
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0answers
33 views
Problems sampling from a $pdf$ over $SO\left(3\right)$
I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series:
$$ f\left(\omega,\theta,\phi\right)=\sum ...
1
vote
1answer
66 views
How do I divide a set of data samples which follow a logarithmic distribution?
I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the ...
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2answers
22 views
How to find density function?
$X \sim N(1,4)$ and $Y = 3 - 5X$. How to find the density function of $Y$?
I tried first to find the distribution function of Y, but got stuck.
$$F(y) = P(Y <= y) = P(3 - 5X <= y) = P(X >= ...
2
votes
1answer
30 views
Range of the distribution of $(1-X)$ when $X$ follows Beta distribution as $X\sim beta(p,q)$
if $X$ follows beta distribution with parameter $p$ and $q$ where $p>0\quad , q>0$
then $1-X$ follows beta distribution with parameters $q$ and $p$,
that is if $X\sim beta(p,q)$ then ...
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1answer
34 views
Probability exponential distribution.
May I please borrow your expertise or could anyone check if I'm on the right track please?
Consider customers arriving at a bank. The bank has $2$ types of customers - business and personal. On ...
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2answers
22 views
Closed form for Exponential Conditional Expected Value & Variance
I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution.
For example:
$$ E(X|x > a) $$ where X is exponential with mean ...
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0answers
21 views
$\mathrm{E}[\log (1 + a X)]$ for non-central chi-squared distributed $X$
I'm really not great in analysis, so I recently got stuck on this problem (Please correct me if I'm wrong somewhere):
Let $Y \sim \mathcal{N}(\mu, 1)$ be a random variable with mean $\mu$ and normal ...
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0answers
48 views
Probability exponential distribution question
Could anyone please help me answer these questions? Or a little hint as to how I can answer them? It's for my assignment that's due tomorrow.. Really appreciate if anyone could help!
Consider ...
1
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1answer
32 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
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1answer
41 views
Boltzmann Distribution With Constraints
I have a problem with showing the existence of Boltzmann distribution given some constraints.
Consider $p_1,...,p_n$ a Boltzmann distibution, where $p_i=\frac{\epsilon^{-\beta \cdot E_i}}{\sum_{j}^{} ...
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vote
2answers
30 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
18 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 ...
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3answers
93 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
13 views
Gaussian expectation of an exponential function
I am struggling to prove this,
$$
\int \mathcal{N}_\mathbf{x}(\mu,\Sigma)e^{a^T\mathbf{x}}d\mathbf{x} = e^{{a^T\mu}+\frac 12a^T\Sigma a}
$$
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votes
1answer
17 views
expectation of logarithm under generalised inverse gaussian
I want to follow the following integral:
$$\frac{1}{C}\int_0^\infty \log(z)\,z^{p-1}\exp\left(-\frac{az+b/z}{2}\right)\,dz$$
where C is the normalising constant.
The following might be useful ...
1
vote
1answer
29 views
Fisher information of a Binomial distribution
The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
1
vote
1answer
25 views
Approximating a Poisson distribution to a Normal distribution
I have the following problem I'm trying to solve:
I know that the quantity of complains in a call center is a Poisson variable with $\lambda=18 $ costumers/hour, and that the probability of being ...
2
votes
1answer
23 views
Fast generation of Pareto-distributed randoms.
I'm developing a library of routines for generating random numbers for simulations (it's on GitHub). I've included fast routines for normally distributed and exponentially distributed randoms, using ...
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0answers
21 views
approximation of Poisson Process witch central limit theorem
I have a conceptual doubt. I am aware of the fact that, if I have to solve a problem including a big summation of identically distributed and independent variables, it is possible to approximate it to ...
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0answers
26 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 ...
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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 ...
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2answers
76 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 ...
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0answers
20 views
please show that $\hat\mu_i\sim N(\mu_i,\frac {\sigma^2}{n_i})$
Statistical model for Complete Randomized design
$y_{ij} = \mu + \tau_i + \epsilon_{ij}$
where, $i$ denotes treatment and $j$ denotes observation.
$i=1,2,...,k\quad and \quad j=1,2,..., n_i$
...
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1answer
36 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) ≥ ...
1
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1answer
34 views
Why is a CDF right-continuous at “a” in [a,b), when property Pr(a<X≤b) doesn't even require point “a” to exist, and “b” could carry baggage?
c.f. wikipedia:Cumulative distribution function properties
"Every cumulative distribution function F is (not necessarily strictly) monotone non-decreasing (see monotone increasing) and ...
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0answers
23 views
Cumulative distribution function of a function of random variable
Random variable $G$ has pdf $f(g)=\frac{2}{3}\cdot e(-2/3g)$ for $g>0$ and $f(g)=0$ otherwise. Now, $L=7$ if $G<5$ and $L=3G$ if G>=5. How to find cumulative distribution function of $L$
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2answers
26 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 ...
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
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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 ...
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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 ...
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1answer
47 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 ...
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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!
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0answers
24 views
Chord length Distribution Function (CDF) in an inhomogeneous medium
Presumably this is a classic problem, but I would need an informed (but nevertheless elementary) answer or citation to start:
Assume a collection of hard, impenetrable 3-spheres (phase B), randomly ...
1
vote
1answer
50 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 $X_1,...,X_n$ ...
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1answer
18 views
please prove the following proof related to F distribution.
Suppose $S_1^2$ and $S_2^2$ are two independent unbiased estimate of the common population variance $\sigma^2$ from two random sample of sizes $n_1$ and $n_2$ respectively.
Then show that
...
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1answer
50 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 ...
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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 ...
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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 ...
