A friend of mine provided me some deep feedback on this article, so I wanted to share it here publicly for posterity purposes with my updated learnings.
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> Unfortunately, your "fun fact" isn't entirely true!
> > "Fun fact: The probability of a Continuous Random Variable (CRV) taking on any exact value is 0."
> In a Dirac distribution, P(x) is 0 everywhere except at one location, where the pdf is such that the integral comes out to 1. So P(y) is 1 for that value.
I learnt that the Dirac delta function is used to represent point impulses. It represents an infinitely narrow spike of probability density at a single point (equaling to 1) and is zero everywhere else. It's integral over the entire real line is equal to 1.
> Bernoulli vs Binomial is such an unfortunate linguistic situation. Such similar concepts. Such similar names. One of the names is semantically unrelated to the meaning of the terms. The other maps closely to the meaning, but is more suggestive of the wrong meaning imo. Bernoulli == single coin flip; Binomial == counting heads.
To reiterate the main point here:
* Bernoulli == single coin flip
* Binomial == counting heads
> Further confused by using dice-rolling as an example -- you do it correctly, but it's suggestive of a six-outcome event whereas we're concerned with just 2-outcome events.
* Bernoulli == single die role
* Binomial == counting number of times a 3 is rolled versus other ([1,2,4,5,6])
> Negative Binomial Distribution: that was a new term for me
Glad we're all always learning :)
> Silly section titles 👌
> Duck probabilities
🦆🦆🦆
> It looks like you just read off the Poisson PDF (0.25)
> Instead, you'd need to choose a range (e.g. 1-2 days) and integrate the pdf over this range (or evaluate the cdf at start and end of range, and subtract end-start). As you noted in your fun fact earlier, the probability of it being exactly one day between duck arrivals is 0!
Even though I do have a CDF, I confused myself because of the numbers I selected and how I defined them, so I'm going redefined the problem here.
* What is the event? Let's assume a duck walking into my house is the event.
* Lambda = 0.5 => time between events is "half a day". A duck walks into my house every half a day, on average. Alternatively, two ducks walk into my house every day, on average.
* Question #1: What is the probability of there being a whole day where a duck doesn't walk into my house?
1 - CDF(P(X >= 1)) ~= 1 - 0.9 = 0.1 = 10%
* Question #2: What is the probability of a duck coming into my house in 6 hours or less (i.e. a quarter of a day)?
CDF(P(X <= 0.25)) ~= 0.3 = 30%
* Question #3: What is the probability of 4 ducks walking into my house in 1 day?
It would be the answer to question #2 to the power of 4
> ChatGPT usage.
> Very cool 🙂
🤖 It is my friend, partner, tutor, assistant and so much more 🤖
A friend of mine provided me some deep feedback on this article, so I wanted to share it here publicly for posterity purposes with my updated learnings.
---
> Unfortunately, your "fun fact" isn't entirely true!
> > "Fun fact: The probability of a Continuous Random Variable (CRV) taking on any exact value is 0."
> In a Dirac distribution, P(x) is 0 everywhere except at one location, where the pdf is such that the integral comes out to 1. So P(y) is 1 for that value.
I learnt that the Dirac delta function is used to represent point impulses. It represents an infinitely narrow spike of probability density at a single point (equaling to 1) and is zero everywhere else. It's integral over the entire real line is equal to 1.
> Bernoulli vs Binomial is such an unfortunate linguistic situation. Such similar concepts. Such similar names. One of the names is semantically unrelated to the meaning of the terms. The other maps closely to the meaning, but is more suggestive of the wrong meaning imo. Bernoulli == single coin flip; Binomial == counting heads.
To reiterate the main point here:
* Bernoulli == single coin flip
* Binomial == counting heads
> Further confused by using dice-rolling as an example -- you do it correctly, but it's suggestive of a six-outcome event whereas we're concerned with just 2-outcome events.
* Bernoulli == single die role
* Binomial == counting number of times a 3 is rolled versus other ([1,2,4,5,6])
> Negative Binomial Distribution: that was a new term for me
Glad we're all always learning :)
> Silly section titles 👌
> Duck probabilities
🦆🦆🦆
> It looks like you just read off the Poisson PDF (0.25)
> Instead, you'd need to choose a range (e.g. 1-2 days) and integrate the pdf over this range (or evaluate the cdf at start and end of range, and subtract end-start). As you noted in your fun fact earlier, the probability of it being exactly one day between duck arrivals is 0!
Even though I do have a CDF, I confused myself because of the numbers I selected and how I defined them, so I'm going redefined the problem here.
* What is the event? Let's assume a duck walking into my house is the event.
* Lambda = 0.5 => time between events is "half a day". A duck walks into my house every half a day, on average. Alternatively, two ducks walk into my house every day, on average.
* Question #1: What is the probability of there being a whole day where a duck doesn't walk into my house?
1 - CDF(P(X >= 1)) ~= 1 - 0.9 = 0.1 = 10%
* Question #2: What is the probability of a duck coming into my house in 6 hours or less (i.e. a quarter of a day)?
CDF(P(X <= 0.25)) ~= 0.3 = 30%
* Question #3: What is the probability of 4 ducks walking into my house in 1 day?
It would be the answer to question #2 to the power of 4
> ChatGPT usage.
> Very cool 🙂
🤖 It is my friend, partner, tutor, assistant and so much more 🤖