Panic! At the lab
Ever have a day where nothing went right? We had a series of unfortunate events so we couldn’t actually do the research we wanted to do. It’s not a huge deal, but what was impressive was the amount of things that went wrong, each thing compounded and let us continue the study up until the point where we were about to start. Only then did we find out we had to stop. There was no danger and the issue was figured out after the fact, but it was an impressive feat of events.
(more…)A fair coin in statistics
It has been a busy day, but the show must go on so to speak and I’m here today to tell you I have a nice shiny new coin for us to flip! The catch you may ask? Well it could be a fair coin or I may have just swapped it out for a coin that was not fair! Our goal is to see if I’m being sneaky and to do that we’re going to need some statistics!
(more…)Day 62: Two Random Variables A2
Like we did with question 1, this will be the solution to the question we posed in the last post, if you haven’t tried to solve it yet, go give it a shot. If you have and are dying to check your answer, then let’s look at the solution.*
Day 61: Two Random Variables Q2
For those of you who have been following along, today we are going to post another question and in the next post we will give the solution. This will be another two random variable question and we’ve covered everything you need to solve it in our previous posts. So with that, let’s get to today’s question.*
Day 60: Two Random Variables A1
Hopefully if you’re reading this you saw our last post, where we gave the question we will solve today. If you haven’t had a chance to try and solve it, please feel free to stop and give it a shot. If you’re ready to see how we solve it, then let’s get started.*
Day 59: Two Random Variables Q1
Well now that we’ve had a minute to take a breath, let’s try out something new. In this post I will give the question and in the next post we can work out the answer. For those of you playing at home, this will be a good way to check your knowledge and for me, it will give the the chance to do the same.*
Day 56: One Function of Two Random Variables Example 3
Today we are solving this guy!
Okay quick example, still not super difficult, but one we can work out to a complete solution. We’ve gone over a few examples now, but we’re going to go over a few more for both my benefit and yours. So let’s dive in.*
Day 55: One Function of Two Random Variables Example 2
Don’t worry, the image ties into what we are doing today, I promise!
Well our last post we took a break and talked zombies! While I would love to do a whole month of halloween topics, this year is not the time, maybe next year. In any case today we are going to go over another example of a single function of two random variables. This is going to be slightly more complex than our first example, however it won’t be extremely complex (we’re working towards it). So let’s take a look shall we?*
Day 53: One Function of Two Random Variables Example 1
Hopefully at this point we’ve demystified more than just a few concepts at this point. Today we are going to look at one function of two random variables. Originally I was going to break into a joint CDF example that involved dependent variables, but it turns out my book doesn’t cover that! Oops, guess I should’ve read ahead. In any case let’s talk functions!*
Day 52: Joint Cumulative Distribution Function Example 1
We’re getting into some interesting things today!
Well here we are again, today we are talking functions of two random variables. If you’re looking for the beginning, this isn’t it, but you can read the introduction here. If you’ve kept up, then you’re ready to go over the example we have today, so let’s get started.*
Day 51: Joint Cumulative Distribution Function
As promised, today we are going to talk about two random variables that are not independent. This means that the individual probabilities don’t sum to be equal to the joint probability (like they did yesterday). Like our normal CDF, we can find a CDF for two random variables, but let’s take a look at how this works.*
Day 50: Intro to Two Random Variables
I was debating about not posting anything today. It’s been a bit rough for me these past few days. However, I’m going to write a little something today and tomorrow to introduce two random variables (so we don’t skip a day). This is going to be a lot like our single random variable examples, but (of course) more complex, let’s take a look at what I mean.*
Day 49: Functions of One Random Variable Example Part 5
Let’s look at something a bit different today!
If you’ve been following along, we’ve been taking it easy the past few days. I’ve also been going over (relatively) simple examples, so today we’re going to take a more complicated example and work through solving it. I may go over a few more examples to be honest, only because this is important to what comes next, after we nail this down, we’re going to get into functions of two random variables and as you can imagine that’s a little more complicated (a lot more).*
Day 48: Functions of One Random Variable Example Part 4
A simpler example than yesterday.
Still feeling under the weather, but the show must go on as they say. So today let’s look at another example of how to solve functions of one random variable. Today will be another shorter post, but as usual I’ll try to explain everything step by step and hopefully you’ll see the methodology behind solving these types of problems.*
Day 47: Functions of One Random Variable Example Part 3
Let’s solve this guy.
Bad news readers, I’m feeling a bit under the weather. Today we’re going to go over a quick example. It’s not the one I had in mind, but it’s fairly straightforward and should help make some more sense of everything we are covering. When I start feeling better we can get into some of the newer concepts.*
Day 46: Functions of One Random Variable Example Part 2
Another Joke by xkcd
As promised today we are going to look at another example of how to solve functions with one random variable. If you haven’t you may want to start with our first post on the subject here. For those of you who are caught up, let’s get started.*
Day 45: Moment Generating Functions
There wasn’t a good visual for what we are talking about today, so how about a joke instead?
Well from the views it looks like my function of a random variable example was a big hit. Hopefully we’re clearing a lot of the confusion up now that we are looking at some examples of how to solve these and we can get a better grasp of how this works. Today we’re going to introduce a new concept, this one is going to be a bit more complex than our last, so you may want to review our last post before we start.*
Day 44: Functions of One Random Variable Example Part 1
Really, this is all we are doing, but let’s make some sense of it.
Well maybe yesterday was confusing, maybe it wasn’t. In any case, today should clarify some things for you if you are confused and should make things more clear if you are not. Today we are going to go over a quick example of what a function involving one random variable looks like. Now you may notice I keep saying one, that’s because you can technically have as many variables as you want, but since this is fairy complex stuff, let’s just stick with the one for now.*
Day 43: Introduction to Functions of One Random Variable
Yep we’re getting a little complicated, but let’s see if we can make sense of this.
Now that we’ve looked at conditional probabilities we can talk about other things we can do with random variables. If you’ve been keeping up with us so far, then this shouldn’t be too crazy of an idea, really all we are going to do today is take a random variable and transform it somehow. Interested? Let’s go!*
Day 42: Conditional Probability
How does this not exist on the internet?! This is directly from my book, so it looks a little… well loved.
Up to now we’ve been dealing with single variable pdf and the corresponding CDF. We said that these probabilities relied on the fact that our variable of interest was independent. However, what if we knew some property that impacted our probability? Today we are talking conditional probability and that is the question we will be answering. It’s going to be a long, long post so plan accordingly.*
Day 41: Connecting the Concepts
Maybe we shouldn’t phrase it this way, since there is still quite a few days left of 365DoA, but you made it to the end! No, not THE end, but if you’ve been following along the past few posts we’ve introduced several seemingly disparate concepts and said, “don’t worry they are related,” without telling you how. Well today, like a magician showing you how to pull a rabbit from a hat, let’s connect the dots and explain why we introduced all those concepts!*
Day 40: The Normal Approximation (Poisson)
Poisson’s return!
Day 39: The Normal Approximation (De Moivre-Laplace)
The binomial distribution, don’t worry we’ll get into it.
Day 38: The Poisson Distribution
The Poisson distribution changes shape as λ changes
Day 37: Bayes’ Theorem
We’re talking Bayes for days.
Day 36: The uniform pdf
The uniform distribution, a very simple looking distribution indeed.
Day 35: Example of the Gaussian pdf
The gaussian (or normal) distribution demonstrated by plinko.
Day 34: Example of the Laplace pdf
The Laplace pdf (left) and the associated Laplace CDF (right). Remember the CDF is just the area under the curve of the pdf.
Day 33: Example of the Exponential pdf
For those who need a refresher, this is a plot of the exponential pdf we are working with today.
Over the past couple of days, I’ve been talking about several different types of pdf and the associated C.D.F. Hopefully, we have a clear understanding of each of those concepts, for those of you scratching your head, I would recommend you start here at this other post. Otherwise, let’s (finally) look at a real life example using the exponential pdf!*
Day 32: The Laplace pdf
The laplace p.d.f with a θ = 0.
Well here we are again… maybe unless you’re new, in which case welcome. If you are just joining us we are talking p.d.f. no not the file format, the probability density function version. If you’re new, you may want to start back here(ish)Â If not, then let’s talk the strangely similar laplace distribution.*
Day 31: The Exponential pdf
Today we are talking this guy! The exponential p.d.f and its C.D.F.
Well, it has been a week, don’t even get me started. But if you’re here you don’t want to hear me complain about my week, that isn’t why we come together! Well today let’s do a bit of a dive into the exponential p.d.f. I hope you’ve brushed up, because this is going to get interesting.*
Day 30: Confidence Interval
Yep, we’re talking confidence!
Day 30 already! Where does the time go? It feels like we just started this whole project and it probably wouldn’t be a good idea to look at the remaining time to completion, so let’s not and just enjoy the nice round 30. We will get back to our p.d.f another day, but today is going to be short. That’s what I usually say before typing out 10 pages worth of information so to avoid that, let’s touch on something important, but something I can do briefly. Today we’re talking about confidence intervals*
Day 29: Probability density functions, Part 3
Don’t be scared, we’re going to tackle this guy today!
Well, apparently you guys really appreciated my probability density function posts. It’s good to see people interested in something a little less well-known (at least to me). So for those of you just joining us, you’ll want to start at part 1 here. For those of you who are keeping up with the posts, let’s review and then look at specific functions. Namely let’s start by going back to our gaussian distribution function and talk about what’s going on with that whole mess. It will be fun, so let’s do it!*
Day 28: Cumulative Distribution Functions
An example C.D.F. of an exponential distribution
Today we were going to do another deep dive into the p.d.f and C.D.F. relationship. Specifically today we were going to talk about specific p.d.f. functions and why we use them, however… I am not doing so hot today, so instead we are going to back track just a bit and talk about what how a C.D.F. differs from our p.d.f. even though we kind of covered it, it would be nice to be clear and I can do this in a (fairly) short post for the day. So that said, let’s get started and we will pick up our p.d.f. discussion next time (maybe).*
Day 27: Probability density functions, Part 2
Today we are looking at our p.d.f. (yes this image has p.d.f. written as PDF, please don’t be confused!) and our C.D.F.’s let’s do this!
Oh hi didn’t see you there. Today is part 2 of the probability density functions notes (posts?), whatever we are calling these. You can read part 1 here as you should probably be familiar with the (super confusing) notation we use to describe our p.d.f. and our C.D.F. now that we’ve given that lovely disclaimer, let’s look once again at probability density functions!*
Day 26: Probability density functions, Part 1
Dashing dreams one comic at a time, via Saturday Morning Breakfast CerealÂ
We are well on our way to wrapping up week 4, what a ride it’s been! It’s been a long day for me, so today might be short. However, I really, really, really want to break into probability density functions. This topic is going to be a bit more advanced than some of the things we’ve covered (IE more writing) so it will most definitely be broken up. Let’s look at why and discover the wonderful weirdness of probability density functions!*
Day 25: The p-value
It’s true!
Now it seems like we are getting somewhere. Last post we covered z-score and you can read that if you haven’t already, it might be good to familiarize yourself with it since today we are going to talk p-value and the difference between z-score and p-value. That said, let’s dive in and look at the value in the p-value.*
Day 24: The z-score
So if you recall from last post… well I’m not linking to it. It was hellishly personal and frankly I’m still attempting to recover from it. We’re going to take it light this time and we can do a deep dive into something in another post. For that reason, let’s talk about z-score and what exactly it is, I mean we used it in this post and never defined it formally, so let’s do that. Let’s talk z-score!*
Day 22: Parametric vs. NonParametric Statistics
Technically we could call this parametric statistics part 2. However, since we are covering nonparametric statistics and more importantly the difference between parametric and nonparametric statistics, it would seem that this title makes more sense. As usual with a continuation, you probably want to start at the beginning where we define parametric statistics. Ready to get started?*
Day 21: Defining Parametric Statistics
It’s halloween time, we are talking about normally distributed data, so this fits, and I don’t want to hear otherwise!
Well my lovely readers, we’ve made it to the three week mark, 5.7% of the way through! Okay maybe that doesn’t seem like a big deal written like that, but hey it’s progress. So last post we had our independence day, or rather defined what it meant to have independent events vs. dependent events. We also said it was an important assumption in parametric statistics that our events are independent, but then we realized we never defined what parametric statistics even is, oops. So let’s stop dragging our feet and talk parametric statistics!*
Day 20: Independent Events
By: xkcd
Because we introduced the central limit theorem last post, it’s time to introduce another important concept. The idea of independent events, while this may seem intuitive, it is one of the assumptions we make in parametric statistics, another concept we will define, but for now let’s jump into independence.*
Day 19: The Central Limit theorem
Well here we are again, if you recall from our last post, we talked Bonferroni Correction. You may also recall that when the post concluded, there was no real topic for today. Well after some ruminating, before we jump into more statistics, we should talk about the central limit theorem. So let’s do a quick dive into what that is and why you should know it!*
Day 18: The Bonferroni Correction
By now we are masters of statistics… right? Okay, not really, but we are getting there. So far we’ve covered two types of errors, type 1 which you can read about here, and type 2 which you can read about here. Armed with this new knowledge we can break into a way to correct for type 1 errors that come about from multiple comparisons. Sound confusing? Well, not for long, let’s break it down and talk Bonferroni.*
Day 17: Type 2 errors
Last post we did a quick bit on type 1 errors. As with anything, there is more than one way to make an error. Today we are talking type 2 errors! They are related in the sense and we’ll go over what that means and compare the two right… now!*
Day 16: Type 1 errors
We did it, we cracked the coin conundrum! We managed the money mystery! We checked the change charade! We … well you get the idea. Last post we (finally) determined if our coin was bias or not. Don’t worry, I won’t spoil it for you if you haven’t read it yet. I actually enjoyed working through a completely made up problem, so if you haven’t read it, you really should. Today we’re going to talk dogs, you’ll see what I mean, so let’s dive in.*
Day 15: Significance, Part 3
Where does our observation fall on the probability density function?
It looks like we’ve arrived at part 3 of what is now officially a trilogy of posts on statistical significance. There is so much more to say I don’t want to quite call this the conclusion. Instead, let’s give a quick review of where we left off and we can get back to determining if an observed value is significant.*
Day 14: Significance, Part 2
Z-score bar graph that I made just for all of you using some data I had laying around. If you’re new to statistics it may not make sense, but rest assured we will make sense of it all!
Well here we are two weeks into 365DoA, I was excited until I realized that puts us at 3.8356% of the way done. So if you remember from last post we’ve started our significance talk, as in what does it mean to have a value that is significant, what does that mean exactly, and how to do we find out? Today is the day I finally break, we’re going to have to do some math. Despite my best efforts I don’t think we can finish the significance discussion without it and still manage to make sense. With that, let’s just dive in.*
Lunatic Laboratories 