Exponential distribution Puns

I recently learned the exponential distribution where P(x)=λe^(-λx), but I’m having trouble with what the probability represents.

Here’s my example problem: Suppose that the length of a phone call, in minutes, is an exponential random variable with decay parameter = 1. If another person arrives at a public telephone just before you, find the probability that you will have to wait exactly 30 seconds. Let X = the length of a phone call, in minutes.

I understand how this works but what would the probability of x=0 represent? The probability that you have to wait exactly 0 seconds? How is this probability equal to 1? Also, why wouldn’t the greatest probability occur at x=1?

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?

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

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)

​

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?

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.

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.

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?

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?

Ten years ago, I read that, if asked to graph out the distribution of wealth in the the US, most people drew a linear slope on a 45 degree pitch. Most were okay with that distribution. They were surprised to learn how hockey stick like the distribution actually was. Most weren't okay with the actual distribution when it was graphed out for them.

I'm all for some inequality. Having the option to put yourself in a better position can provide some sense of purpose and something to work toward. A straight slope with some pitch entails more opportunities for more people to make themselves better off. Whereas, a reverse L shaped distribution only leaves room for a limited number of people to improve their circumstances (albeit - drastically).

Technological improvements seem to be making previously local industries nationally scalable and thereby put the profit from those opportunities into the hands of the few.

The equalizing solutions proposed by my government (Canadian) end up targeting the people who are well on the right side of the curve but not in the vertical part of the slope. For example, higher income taxes. Even capital gains taxes and dividend taxes make it harder for those in the horizontal part of the curve to improve their circumstances.

Some kind of wealth tax would seem to make the most sense. Alternatively, one set of tax rules for people in the horizontal part of the curve and another, more expensive, set for those on the vertical part might make sense.

However, in response to those suggestions, I usually read that

• The tax structures of the rich are too complex to apply those solutions to.

• The nature of wealth (liquidity especially) varies too much to tax.

• The wealthy would leave the country with their money if such a tax were enacted.

If that's true, then I can't imagine there's anything we can do. We're just doomed to watch all the wealth and opportunity shift farther to the right of the curve.

Moreover, the 'tax the rich' solutions don't seem that they would do much to create more opportunity anyway. The tax money might be used to provide more education to more people or whatever else - but the fact remains, if we have hockey-stick wealth distribution, then there's only a tiny sliver of people who will end up getting ahead. Providing education and child care (for example) might mean that the at-birth-odds of a person landing in the 99.9th percentile of wealth would be closer to equal, but we would still have the s

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

Let z ~ N(mean,variance) and Let Y = exp(Z)

Find the probability density function of Y

Hello fellow data people. I was recently working with a dataset I had cleaned up and filtered. Machine learning (Tensorflow, custum, Microsoft Azure), trend functions (linear, exponential, etc.), and a bunch of other forecasting techniques had terrible results (yes, I tried using autoregressive inputs). I then plotted the frequency distribution of the outputs of the dataset, only to find out that it is an exponential probability distribution. Basically, there's 8 billion 0s, 3 billion 1s, 1.5 billion 2s (they're count values). I was wondering what useful forecasting can even be done when a dataset looks like this. Honestly, I can't even imagine any useful summary statistics (like, "Oh, great! I have a 90% chance of it being less than 2"). This is like the third dataset I've had that looks like this and I'm always lost when I get them (I've done hackathons in the past that have had this same problem). Any help would be appreciated.

I would like to understand the nature of measure, using the exponential family of probability distributions as a context (because I understand the latter well). I understand that we want equation (8.1) to integrate to 1 (this is the definition of a probability distribution). Therefore we set that integral to one, and take the e^(-A(n)) term out of the integral (since it is not a function of x), and rearrange to get equation (8.2).

My point is, I understand what's going on algebraically, but I have no idea what role the measure is playing conceptually. I have tried to learn measure theory many times (i.e., on my own), but I just don't seem to get what it is or its motivation. When the word measure is used, I have no idea what it refers to. The Lebesgue measure is supposed to be a generalization of length to sets that more complicated than intervals. I understand it somehow relates to probability. But what is a measure? In the particular context below, for example, what does it contribute to the description of the exponential family?

Source: https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/other-readings/chapter8.pdf

https://preview.redd.it/ose94uoav0d51.png?width=673&format=png&auto=webp&s=424d8cd84e7a03db15ea65c39050c07aca475a41

It is very naive question, but I really get confused. As I saw in some plots the maximum of the the probability density function explained by the exponential distribution function has a maximum at a non zero parameter, as in the formula we have the f(x)=lambdae^(-lambdax). And the maximum will be at the x=0, how in some distributions their maximum as at some other positive values of x?

http://imgur.com/gallery/TxulT6S

I did the easiest one.

Someone else already done normal(or binomial).

Who will go for a harder one like log-normal or student-t?

The Picture

I reran with power and exponential decays. I dont like how they vertically shoot up from nowhere, but other people thought them interesting. Ive also drawn histograms of all the distributions Ive tried.

I wanted to try some smoother exponentials, but to keep the mean theyd have required some people to have negative connectedness.

Table form:
lognorm[0.0] min=2.9 25th=3.0 med=3.0 75th=3.0 max=3.1 turning point=60.3% final=94.1%
lognorm[0.5] min=0.2 25th=1.9 med=2.6 75th=3.7 max=28.5 turning point=69.6% final=88.3%
lognorm[0.8] min=0.1 25th=1.4 med=2.3 75th=3.8 max=80.0 turning point=47.2% final=82.1%
lognorm[1.0] min=0.0 25th=0.9 med=1.8 75th=3.6 max=211.0 turning point=53.1% final=74.7%
lognorm[1.5] min=0.0 25th=0.4 med=1.0 75th=2.7 max=1217.0 turning point=52.9% final=58.4%
exp[2.0] min=1.6 25th=2.0 med=2.6 75th=3.6 max=21.5 turning point=69.0% final=90.0%
exp[1.7] min=1.1 25th=1.7 med=2.4 75th=3.7 max=27.2 turning point=71.2% final=87.1%
exp[1.5] min=0.5 25th=1.2 med=2.2 75th=4.0 max=34.6 turning point=46.1% final=81.5%
power[2.1] min=0.5 25th=0.7 med=0.9 75th=1.4 max=99313.2 turning point=39.7% final=60.0%
power[2.5] min=1.1 25th=1.3 med=1.7 75th=2.6 max=9410.3 turning point=64.7% final=80.4%
power[2.9] min=1.5 25th=1.7 med=2.1 75th=3.0 max=1910.1 turning point=55.2% final=86.8%

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Hi all! I have a super basic question

I'm currently marking some stats exams.

The exponential distribution typically is written with support y>=0.

I have many students writing y>0. Is this also equivalent?

For example, I have set of time series data, and when you plot their distributions, from a glance they all look like semi-normal or exponential distributions.

I'm looking for ways to measure and compare the fat "tailed-ness" of their distributions.

• Is using the mean/median ratio an appropriate way to measure this? I thought it would be as the ratio will capture how much the mean is skewed by the tails.
• What other measures or statistics can I use to measure this?

Does anyone have a picture/webpage where it shows how both the weibull and exponential distribution come from the gamma distribution?

Thanks!

I’m writing about queueing theory for my high school paper, and I’m not very familiar with this subject area as it’s completely out of our syllabus.