Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.
1
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2answers
61 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
2answers
43 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
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1answer
31 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 ...
1
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0answers
17 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
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1answer
32 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) ≥ ...
2
votes
1answer
277 views
How do I calculate the aposteriori probability distribution for someone's answer to a poll being an approval?
Imagine I'm polling a random sample from the population and it asks them if they approve of the President or not. I also ask them some categorical demographic questions (age-bracket, race, gender, ...
0
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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
...
1
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1answer
38 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
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1answer
143 views
Probability regarding supply and demand with Normal Distribution
The question: A sell-out crowd of $42,200$ is expected at Cleveland's Jacobs Field for next Tuesday's game against the Baltimore Orioles, the last before a long road trip. The ballpark's concession ...
0
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0answers
21 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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1answer
139 views
Probability for Communication Networks
A computer communication channel transmits words of $n$ bits using an error-correcting code which is
capable of correcting errors in up to $k$ bits. Here each bit is either a $0$ or a $1$. Assume ...
0
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2answers
24 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
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1answer
28 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 ...
1
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1answer
28 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
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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 ...
0
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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!
0
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0answers
18 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 ...
0
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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 ...
0
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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 ...
0
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1answer
45 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 ...
1
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3answers
41 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
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0answers
15 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 ...
0
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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, ...
0
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0answers
37 views
FF neural network with single sigmoid output (calculation of probability)
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
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1answer
326 views
Correlation between Beta distributions
I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
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.
...
4
votes
1answer
15 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 ...
0
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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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1answer
26 views
Explain why a chi-square random variable will approximately have a normal distribution for large n
Explain why a chi-square random variable having n degrees of freedom will approximately have the distribution of a normal random variable when n is large.
2
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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 ...
0
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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
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0answers
60 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
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1answer
27 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 ...
1
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1answer
31 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
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0answers
9 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
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0answers
14 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 ...
0
votes
1answer
47 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
44 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 ...
4
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0answers
210 views
Simulating from a Multivariate Gaussian without Cholesky
I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
0
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1answer
76 views
Find density for $Z=X+Y$ with joint density function
Find the density function of $Z=X+Y$, $X,Y$ have the joint density function $f(x,y) = \frac{1}{2} (x+y) e^{-(x+y)},\, x,y \geq 0$.
My initial idea is to calculate the distribution function of Z like ...
2
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0answers
17 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 ...
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1answer
52 views
Computing PDF of Products of Two Random Variables
I've been stuck on this problem for some days. I'm hoping someone would help by chipping in a few comments. I have two i.i.d. r.v.:
$$
f_X(x)=\frac{\left(1-e^{-\frac{x}{\alpha }}\right)^{\tilde{r}-1} ...
1
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3answers
43 views
Doubt about why I can't treat this as a Bernoulli process
I know the title is not descriptive enough, but I don't know how else to say it. I don't know why I can't use the Binomial distribution to get the result I'm looking for. The teacher solved it long ...
1
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3answers
26 views
prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$
How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$.
By memoryless property,
$$P(X=i+s | X>i)=P(X=s)$$
How to get ...
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 ...
0
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1answer
23 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
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4answers
42 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 ...
2
votes
2answers
130 views
Relating two proofs of binomial distribution mean
There are two ways of calculating the mean of the binomial distribution. One is to observe that the distribution measures the number of successes in a sample size $n$ drawn from space of size $N$. ...
1
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0answers
24 views
Probability distribution of the product of two independent complex gaussian random variables
I have to calculate the pdf of $Z = X*Y$, where $X \in \mathcal{C}(\mu_x,\Sigma_x)$ and $Y \in \mathcal{C}(\mu_y,\Sigma_y)$, where $\mathcal{C}$ is a complex distribution. It can be assumed that ...
