Negative Exponential Distribution Puns

I think I remember learning that if you randomly populated a 2D plane or 3D space, say with dots, you would get the Normal distribution. Is this correct, and if so how it would the Normal distribution manifest?

And is the same true for the exponential distribution? Would randomly populating a 2D or 3D space with dots, or a 2D or 3D grid with cubes or 1's and 0's, result in the exponential distribution somehow appearing?

And are there any other common distributions that would somehow manifest in these randomly populated space scenarios?

By randomly populated a plane or space, I mean if the plane was like a grid, and you randomly filled X% of the grid with dots. So let's say you have a 10,000 by 10,000 plane, and you randomly filled 2% of it or 10% of it with points. I think I remember learning that the Normal distribution shows up in the distance of the points from each other, or the distribution of the average distance of the points from each other, or something like that.

I wasn't sure whether to flair this with Geometry or Statistics, I picked Geometry, my apologies if Statistics was more appropriate.

https://youtu.be/W2UT8NjUqrk

Backpropagation is the workhorse of deep learning, but unfortunately, it only works for continuous functions that are amenable to the chain rule of differentiation. Since discrete algorithms have no continuous derivative, deep networks with such algorithms as part of them cannot be effectively trained using backpropagation. This paper presents a method to incorporate a large class of algorithms, formulated as discrete exponential family distributions, into deep networks and derives gradient estimates that can easily be used in end-to-end backpropagation. This enables things like combinatorial optimizers to be part of a network's forward propagation natively.

OUTLINE:

0:00 - Intro & Overview

4:25 - Sponsor: Weights & Biases

6:15 - Problem Setup & Contributions

8:50 - Recap: Straight-Through Estimator

13:25 - Encoding the discrete problem as an inner product

19:45 - From algorithm to distribution

23:15 - Substituting the gradient

26:50 - Defining a target distribution

38:30 - Approximating marginals via perturb-and-MAP

45:10 - Entire algorithm recap

56:45 - Github Page & Example

Paper: https://arxiv.org/abs/2106.01798

Code (TF): https://github.com/nec-research/tf-imle

Code (Torch): https://github.com/uclnlp/torch-imle

The way i understand calculating a irration exponent is that you find the limit of what the number approach when you keep adding one and one decimal. for. ex 2^sqrt(2) is the limit of

2^1 2^1.4 2^1.41 2^1412.

the problem if the base is a negativ number is that only half on the numbers will be "solvable". do that mean it's just impossible?

The follow-up question is exponetial function with a negativ base, would that alternate back and forth in the same way between solvable and non-solvable? Yet in wolfram alfa it seems to just drift between negativ and positive. like this: https://www.wolframalpha.com/input/?i=%28-1.1%29%5Ex

Can you guys help me out, everything i find on google is just focuse on much easier math...

https://imgur.com/uYJjCRv

One of the questions asked us to find the MLE of this distribution, which I calculated to be 1/(Xbar - 2). I think this is correct because I checked with someone else. However, the next question asked us to determine if this MLE is biased. I don't really know how to approach this problem. I tried using the property that a sum of n IID exponential variables, let's call this y, has a gamma distribution, then tried to set up an intergral which used the pdf of this gamma distribution multiplied by n/(y-2n) over 0 to infinity but I can't solve it and I feel like there is an easier way to solve this question?

Hi all!

Today I've been pondering about the geometric and exponential distributions. They're the two main memoryless distributions, while one is discrete and the other continuous (on the positive reals). So I had a pretty simple idea: You can "approximate" an exponential distribution by partitioning the positive reals as follows:

Let $E \tilde Exp(\lambda ) , \lambda > 0$. Choose $\epsilon > 0$ and define $ X = \lbrace i\epsilon \mid i \in \mathbb{N} \rbrace$. Now we can define a geometric distribution on this set. The only thing that remains to be chosen is the transition probability $p$. We choose it, such that we mimic the exponential distribution: We use the simple fact that

$P(E < (i+1)\epsilon \vert E > i\epsilon) = 1 - P(E > (i+1)\epsilon \vert E > i\epsilon) = 1-e^{-\lambda\epsilon}$

Thus we define the Variable $G_{\epsilon}$ according to the geometric distribution $Geo(X,1-e^{-\lambda\epsilon}$ .

Now, is it true that $P(G\in [a,b]) \rightarrow P(E\in [a,b])$ as $\epsilon$ goes to 0? I think it is, it's pretty intuitive this far. At least for $[a,b] = [k\epsilon ,l\epsilon]$ we have

$ P(G \in [k\epsilon , l\epsilon] ) = e^{-\lamba\epsilon l} - e^{\lambda\epsilon (k+1) = F_{E}(\epsilon l) - F_{E}(\epsilon (k+1)) $ where $F_{E}$ is the CDF of $E$.

If so, we can think of a Poisson-Process $P_t$ where the interarrival times are i.i.d. exponentially distributed. Could we define any interesting Stochastic Processes (maybe a Markov-Chain) using $G_{\epsilon}$ that relate in any way to the Poisson-Process (or it's embedded Markov-Chain)?

Thanks in advance!

My experimental results are exponentially distributed. I estimated the exponential parameter as 1/mean, but how should I calculate its uncertainty?

> It’s a tale of two countries. For the 117 million US adults in the bottom half of the income distribution, growth has been non-existent for a generation, while at the top of the ladder it has been extraordinarily strong. And this stagnation of national income accruing at the bottom is not due to population ageing. Quite the contrary: for the bottom half of the working-age population (adults below 65), income has actually fallen. In the bottom half of the distribution, only the income of the elderly is rising.5 From 1980 to 2014, for example, none of the growth in per-adult national income went to the bottom 50%, while 32% went to the middle class (defined as adults between the median and the 90th percentile), 68% to the top 10%, and 36% to the top 1%. An economy that fails to deliver growth for half of its people for an entire generation is bound to generate discontent with the status quo and a rejection of establishment politics.

https://voxeu.org/article/economic-growth-us-tale-two-countries

If a call center is functioning from 8:00 AM to 8:00 PM every day, and incoming calls occurs according to a Poisson process with rate 0.1 per minute, then for any given day, the probability that 8th call occurs between 8:40 AM to 8:50 AM given that the first call occurred at 8:10 AM, is  -

Let's say there is a signal x(t) and it Fourier Transforms to X(f). Now let's say that x(t) is being multiplied by a negative complex exponential [e^(-j*2*pi*f0)]. So now we have x(t)*e^(-j*2*pi*f0). I know that FT{x(t)*e^(j*2*pi*f0)}=X(f-f0). However, I am unsure what happens when the complex exponential is negative. Is it X[f-(-f0)] or is it still just X(f-fo)??

Edit: it should be e^(j*2*pi*f0*t). I forgot the "t"

I have attempted this problem several ways and still remain perplexed. Any insight or assistance would be much appreciated:

Tom the cat patrols the house in search of the mouse Jerry, who wants to get into the kitchen for something to eat.

After leaving the kitchen, Tom will return after a random amount of time, which follows an exponential distribution with mean of 15 seconds. Assume also the time needed for Jerry to cross house is an exponential random variable with mean of 10 seconds.

Since Jerry needs 10 seconds to get across the house, and starts immediately after Tom leaves the kitchen. What is the probability that Jerry the mouse will get caught by Tom the cat?

One of my colleagues fit parametric distribution on survival plot using SAS, I wonder if anything similar can be done with R on KM curve or its derivatives?

Hello, all!

I'm getting very confused regarding the use of the λ parameter used on both Poisson and Exponential distribution.

I've learned that the lambda in Poisson is the average occurrences by a unit of time/space. But when It comes to the exponential distribution, I often see the lambda being said to be its mean, and the E(X) being 1/lambda.

Example from Wikipedia:

https://preview.redd.it/89wyg8bketj61.png?width=722&format=png&auto=webp&s=8253d44c944b645e965626d4e25ec58e8d5d6d28

And then it says that the mean is 1 divided by the rate parameter, which is going to give us the Poisson lambda.

https://preview.redd.it/60ywzc2vetj61.png?width=709&format=png&auto=webp&s=4cf42aa4419556dbf35bdc5c8ff0c2c2e686a048

So I would like to know.

1. What's the real E(X) of an exponential distribution? Is it lambda, from the Poisson distribution or the rate parameter itself?
2. What's the difference between the lambda used in the Poisson distribution and the exponential distribution?

I'm not being able to find other resources that teach this subject in-depth, if you know any other resources I would be very thankful if you point them out.

Hi everyone, I have two questions regarding exponential distribution:

1. The mean or expected value of Exponential Distribution is 1/ λ . However whenever I try proving the same using python it never works out. What am I doing wrong? (Code Below)

&#8203;

import numpy as np
n = 1_000_000
Y = np.random.exponential(5,size=n)
print(np.mean(Y))

>> 5.0004826909044855

Shouldn't the mean of the r.v Y be 1/5 or 0.2 as per the definition and not λ itself**?**

2. According to Joseph Blitzstein Introduction to Probability, if X ~ Exp(1) then Y ~ Exp(λ) if Y=X/λ.
   Shouldn't Y be λ*X to be distributed as Exp(λ)? Even plotting the distribution of of Y looks like
   Exp(1/λ) instead of Exp(λ). What am I missing here?

Hey friends,

I've been stuck on this problem for quite a bit, and could really use some guidance.

The time (in minutes) between arrivals of customers to a post office is to be modelled by the Exponential distribution with mean 0.72.

What is the probability that the time between consecutive customers is less than 15 seconds?

For some reason using exponential distributions in this manner has really been throwing me through a loop, and I'm not entirely sure how to start. I've rewatched my lecture videos, and its just not clicking. Does anyone have some extra resources that could help, or guidance on the problem? Personally, I would prefer to understand the mechanics of the topic, because I would really like to move on.

Hello all,

I am wondering if it is possible to move exponential distribution to a certain interval. If it is possible to move probability density function to a certain interval?

Dear community,

I have the following exercies: https://imgur.com/a/1rvAJU2

The solution is highlighted. I did 1-e^(-1/1.42)*4 and got result b) (integral from 0 to 4)

What did I do wrong?

I have an exponential function that I want to generate random numbers from. It is like this:

f(x) = 1 - 1.01 * exp(-x/2.97)

I thought of using the numpy.random.exponential function. But it only allowes the use of functions of the format:

f(x) = 1/b * exp (-x/b)

Is there any numpy or scipy routine that will allow me to draw random numbers from a user defined function?

Hello, currently a undergrad stats major at my university, in a probability class. We just learned a whole host of probability distributions today, gamma, exponential, beta, chi square. And I came here to ask a question to clarify the relationship between them.

To my understanding and the way I was thinking about it was.

Gamma distribution, is skewed and takes up values which are in the set of Reals (0 to infinity). Only positive values. Has its respective variance and expected value

Exponential, chi square, and beta are all TYPES of a gamma distribution, and that a gamma distribution is like a overarching baseline for the three of them. But each had minor differences.

Exponential is parameterized strictly with a = 1 and B. So B will be most likely to change but alpha will always stay. Is also from 0 to infinity. Had a memory less property (don’t really understand why it’s for this and not others)

Beta, is strictly over the interval 0 to 1, parametized by alpha and beta

Chi-square, modeled by a gamma but has some constant v? Dont really get this either.

So could someone maybe fill in the gaps in my understanding here? Specifically

How do the 4 relate to each other?

What is the memory less properly in relation to the exponential distribution?

Why does chi square have this letter “v”?

Thanks

If I have a variable and know the mean and standard value, there's a probability, albeit small, that the actual value could be way out in the tails.

But what if I'm measuring the time something will take? Clearly it will not take negative time, and so a standard deviation does not make sense. What is the correct distribution?

When I've asked this question in the past, people have suggested a bunch of non-negative distributions and told me to "go try those and find one that you like". I don't want to know just any distribution, I want to know what is the distribution I should expect. Like, if I went into a lab and ran the measurement a hundred thousand times, what shape would I see, and what is the mathematical formula for that given an infinite number of trials?

According to this paper: yes.

https://voxeu.org/article/economic-growth-us-tale-two-countries

> It’s a tale of two countries. For the 117 million US adults in the bottom half of the income distribution, growth has been non-existent for a generation, while at the top of the ladder it has been extraordinarily strong. And this stagnation of national income accruing at the bottom is not due to population ageing. Quite the contrary: for the bottom half of the working-age population (adults below 65), income has actually fallen. In the bottom half of the distribution, only the income of the elderly is rising.5 From 1980 to 2014, for example, none of the growth in per-adult national income went to the bottom 50%, while 32% went to the middle class (defined as adults between the median and the 90th percentile), 68% to the top 10%, and 36% to the top 1%. An economy that fails to deliver growth for half of its people for an entire generation is bound to generate discontent with the status quo and a rejection of establishment politics.

Is this accurate and if so why is this happening?

Also does the negative binomial distribution have the memoryless property like the geometric distribution does?