Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.
1
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1answer
15 views
Calculating the probabilities of different lengths of repetitions of X length numbers
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
38 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}^{} ...
1
vote
2answers
27 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
16 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 ...
2
votes
3answers
86 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
11 views
Gaussian expectation of an exponential function
I am struggling to prove this,
$$
\int \mathcal{N}_\mathbf{x}(\mu,\Sigma)e^{a^T\mathbf{x}}d\mathbf{x} = e^{{a^T\mu}+\frac 12a^T\Sigma a}
$$
1
vote
1answer
24 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
21 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 ...
0
votes
0answers
20 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 ...
0
votes
0answers
23 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
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
vote
2answers
72 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 ...
1
vote
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.
$i=1,2,...,k\quad and \quad j=1,2,..., n_i$
...
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) ≥ ...
1
vote
1answer
33 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 ...
0
votes
0answers
22 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 $L$
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 ...
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
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
0answers
21 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
44 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!
0
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0answers
22 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
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 $X_1,...,X_n$ ...
0
votes
1answer
17 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
...
0
votes
1answer
47 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 ...
1
vote
3answers
42 views
Poisson Distribution - sum of RVs
Question:
$X$ balls are thrown to $n$ bins (each ball has an equal chance to get to each bin). Let $X_1,\dots, X_n$ be the amount of balls in each cell.
a. Show that if $X \sim ...
1
vote
1answer
19 views
How Entropy scales with sample size
For a discrete probability distribution, the entropy is defined as:
$$H(p) = \sum_i p(x_i) \log(p(x_i))$$
I'm trying to use the entropy as a measure of how "flat / noisy" vs. "peaked" a distribution ...
1
vote
0answers
55 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 ...
0
votes
1answer
22 views
Please help finishing the calculation to find the Entropy of Pareto distribution.
Let $X$ follow Pareto distribution with parameters $\alpha, a, h$. That is, $X\sim Pa(\alpha,a,h)$, where $\alpha>0$ is the shape parameter, $-\infty < a < \infty$ is the location parameter, ...
4
votes
1answer
22 views
How does the increase in overall number of events affect the peak (events/time)?
I have a (simple?) question that I hope someone will find interesting enough to help me out with.
A web site has a given number of subscribers who generate a certain amount of traffic on the web ...
1
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0answers
26 views
Please help finishing the calculation to prove that ” Pareto distribution & Power distribution has inverse relationship”.
Let X follows Pareto distribution with parameters α, a, h.
that is X~Pa(α,a,h)
Where, α>0 is the shape parameter, -∞< a <∞ is the location parameter, h>0 is the scale parameter.
...
0
votes
1answer
28 views
A Measure For The Space of Probability Density Functions
Consider the space of all joint probability density functions of two variables. I want to know what the measure is of the portion of this space that is filled by uncorrelated joint pdfs relative to ...
4
votes
0answers
62 views
Determining a confidence interval for $\sigma$ from a Rayleigh distribution
Hello stackexchangers,
Suppose we have $n$ Rayleigh distributions defined by
$$f_X(x)=\frac{x}{\sigma^2}e^{-x^2/2\sigma^2}.$$
How would you go about determining an approximative confidence interval ...
1
vote
1answer
29 views
Is $\left(X_1,… ,X_n,\bar{X}\right)$ jointly normal distributed if $\left(X_1,… ,X_n\right)$ is?
Let $X:=\left(X_1,... ,X_n\right)\sim N_n(\mu,\Sigma)$, $\mu\in\mathbb{R}^n$, $\Sigma\in\mathbb{R}^{n\times n}$ symmetric and positive semi-definite and $\bar{X}:=\frac{1}{n}\sum_{i=1}^n X_i$ as ...
0
votes
0answers
10 views
Distribution of partial sums of a $L^2$-transformed Gaussian Process
Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
1
vote
0answers
15 views
Central Limit Theorem for Dependent Non-Identical Random Variables.
If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$.
How do we apply ...
2
votes
1answer
45 views
Sum of Bernoulli random variables with different success probabilities
Let $X_{i} \in \{0,1\}$ be Bernouli random variable with probability of success $p_{i}$, i.e., $P(X_{i}=1) = p_{i}$ and $P(X_{i}=0) = 1-p_{i}$ and let $Y=\sum_{i=1}^{n}X_{i}$ for $n>0$. Is it ...
1
vote
1answer
33 views
Constructing Distribution By Coin Flipping
I am interested in any example of construction distribution by coin flipping.
Actually I want to show the process of construction a random variable $X$ with distribution $(p_1,...,p_n)$ by coin ...
0
votes
1answer
48 views
Proof that a sequence of random variables have finite expectation
Let $X_n$ be iid non-negatives random variables. Prove that $\mathbb{E}[X_1] < \infty$ iff $P(X_n \ge n\text{ i.o.}) = 0$
I thought I would start like this for one direction
$\infty > ...
2
votes
1answer
18 views
Basic understanding of sampling from a continuous distribution.
For continuous distribution (on R) the probability of a single point is $0$.
So I'm not sure what does it mean to sample $M$ elements from a continuous distribution.
Let say there is a continuous ...
2
votes
0answers
19 views
Inferring a probability distribution from another probability distribution
Let $A$ and $B$ be real-valued random variables, with $f_A$ and $f_B$ their probability density functions. Let's say we can observe the values of $A$ many times and estimate $f_A$ fairly precisely. We ...
1
vote
3answers
44 views
Doubt about why I can't treat this as a Bernoulli process
I know the title is not descriptive enough, but I don't know how else to say it. I don't know why I can't use the Binomial distribution to get the result I'm looking for. The teacher solved it long ...
1
vote
3answers
27 views
prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$
How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$.
By memoryless property,
$$P(X=i+s | X>i)=P(X=s)$$
How to get ...
0
votes
1answer
25 views
Integrate over the uniform distribution on the simplex
Let $p=(p_1,\ldots,p_n)$ correspond to points in a simplex that add up to one, i.e. $p$ is a discrete probability distribution. I would like to compute an integral of the form $\int dp_1\ldots\int ...
0
votes
4answers
43 views
A basic doubt on the definition of a Poisson random variable
What is the significance of "large city" in the definition of the following Poisson variable :
"Number of phone calls placed during a ten second interval in a large city"
I guess either $n \to ...
0
votes
2answers
29 views
Expectation Values
Suppose that $\{X_n\}_{n\ge1}$ take values $-\dfrac{1}{2}$ and $\dfrac{1}{2}$ with probability $a$ and $1-a$ respectively and $0$ otherwise. Suppose further that they are independent and discrete and ...
1
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0answers
22 views
limit distribution of possion distribution
Assume Xn is possion with mean $\lambda_n$ and suppose that $\lambda_n\rightarrow\infty$ as $n\rightarrow \infty$. Then how to show Xn is AN$(\lambda_n,\lambda_n)$. I've tried to use characteristic ...
