Probability Density Function Puns

Really the title speaks for itself. I’m taking quantum and this was taught to us when Schrödinger was discussed. The problem is I can’t get my head around the fact WHY this is true. We weren’t really given any derivation, it was more like a statement that we had to accept lol. Youtube videos explain it the same way, not giving any reason for why it is this way :|

Ok let me start off with the dumb question, I saw this in my notes and I embarrassingly have no idea what they mean.

X<Y<Z, X<=Y<=Z, and few variations of that. I'm sure it’s like X is greater than or equal to Y which is greater than or equal to Z or something like that but that could also be super wrong.

On to the more difficult question, I saw those symbols in a graph about Probability Density Functions.

I literally do not understand it. I know it's about the probabili6y of an event happening in an infinite space of time and that the probability can be illustrated with bell curve graphs but I don't really have a more intimate understanding of that. Could someone explain like they were explaining it to a really dumb toddler please and thank you

I’ve tried to read my notes and watch videos that my professor gave but it seems I lack even basic math skills and am really far behind so it all goes over my head

Translation:

'Considering that we have an unlimited bandwidth, we use a VPN to download a file from the Internet via an external IP. The time to download in seconds has the following distribution given by the density function: [function in image]

a) Calculate the probability that the file takes longer than a minute to download.

b) If after a minute the download has not finished, calculate the probability that it takes longer than 2 minutes.

c) If 3 distinct and independent IP addresses are used to download the same file, calculate the probability that it takes less than one minute to download the file (with any one of the IP addresses).'

I thought that for a and b the answer should be the total sample space minus the probability of the file taking a minute to download, and likewise for b.

Because of the wording of b I thought it was asking for a conditional probability, so I tried to visualize it using a venn diagram but I don't think I've defined the events correctly (and I'm not even clear whether you can use venn diagrams for continuous variables...??) How would I write the conditional probability for the situation in b?

C is the part I'm having the most trouble with. Usually if events are independent and you want the total probability that ANY ONE of them occurs, it would be something like A OR B OR C therefore they are added...right? But if, in theory, I could have any number of IP addresses and just kept adding the probability that they take less than a minute to download the probability would just end up being more than 1 eventually so I know I'm looking at this wrong... What do I do here?

Question

My attempt so far...

Hello,

I'm a Maths major and have recently been learning about PDFs and DEs. Learning about PDFs opened up a whole new lens to view the world through, and I subsequently started expressing the output of my garden in terms PDFs. What I'm curious about is the connection between PDFs and DEs, to me they are two sides of the same coin, two different ways of expressing the complicated relationship between variables. I want to know if there is any math that makes the connection between PDFs and DEs more concrete than my conceptual understanding?

Any feedback would be much appreciated.

From what I have read, and understood is that PDF describes cumulation rate of random variables in particular neighborhood of a distribution curve. Are there any differences, in term of formula of PDF, between Gaussian and Binomial probability distribution?

I have googled this but unable to find a satisfied answer.

I have very limited knowledge of statistics, and would appreciate an elaborated answer, or a link to one. Thanks!

I have seen many lectures about this topic, but still can't wrap my head around it. I know the Probability Mass Function (PMF) which is used for mapping discrete and countable probabilities, but how exactly do you map continuous and uncountable probabilities? Please explain with analogy if possible. Thank you.

if i have a function of probability density function and i would like to get probability out of it will multiplying the outcome by the interval suffice?

For example Interval is 0.5... divide by 2 will produce correct answer or am i making a double mistake?

---solved ty

https://imgur.com/a/6bvhuxp

In the solution why did they not sketch for x is less than zero.

thanks

In the textbook from uni I had for probability and Math Stat (Wackerly), we (the textbook) defined various random variables with a definition that went something as follows:

"A random Variable Y is said to have a ________ probability distribution if and only if for parameter _______ and parameter _______ the density function of Y is: Insert complicated function here

This troubled me because it offered very little (usually zero) justification for why that function exists or its derivation.

Where do all these density functions come from? Did statisticians just try to come up with random (no pun intended) functions that integrated to 1 and then hope they had meaning and relevance to our world? (I don't actually think this is the case)

Are there any resources that contain the justification/derivation for the PDFs and CDFs we commonly use?

Are these explanations just too complicated for my feeble mind, and the authors are simply sparing me from inevitable suffering...

Are these two terms interchangeable? The best I’ve been able to come up with to answer this question is this quote from the first line of the wikipedia article on PDFs:

“a probability density function (PDF), or density of a continuous random variable,...”

which seems to imply these are the same, but would appreciate confirmation.

x-posted from /r/stats which seems close to dead.

Hey, I'm a mathematics student.

As of how I currently understand it, the probility density function of a normal distributed random variable doesn't have an anti-derivative. Although I've seen this said in a couple of books I would like to know whether there is a proof for this statement as I haven't come into contact with one yet. Thus I would like to ask whether it can be proven and if so how, if not why not and why do we still treat it as a fact? Thanks alot in advance!

For a continuous variable x and its probability density function p(x), I have a numpy array of x values x and a numpy array of corresponding p(x) values p. p(x) is not normalised though, i.e. in a plot of p(x) against x, the area under the graph is not 1. I want to calculate a corresponding array for values of the cumulative distribution function cdf. This is how I'm currently doing it, using the Trapezoidal rule to approximate an integral:

pnorm = p/np.trapz(p,x)

cdf = np.array([np.trapz(pnorm[:n],x[:n]) for n in range(len(pnorm))])

The results aren't entirely accurate; the final value of cdf is close to 1 but not exactly 1.

Is there any more accurate and simple way of normalising p and finding cdf? I thought there might be specific functions for this in some module; perhaps a statistics-oriented module with functions for related parameters (variance, confindence intervals etc) as well?

Hi,

I want to generate pdfs from caps and floors to gauge the likely path of interest rates over the next couple of years.

I’m looking for some reasonably penetrable reading material with practical examples. Many moons ago I majored in mathematics but safe to say I’m incredibly rusty.

TIA

[Solved] Hello, everybody. Is there a way to mark areas μ, μ−2σ, μ−σ, μ+σ, μ+2σ (μ - mean, σ - standard deviation) in a probability density function graph picture below? How can I select specific value interval and colour a corresponding chart area? The result should look similarly to normal distribution.

Solution.: Vertical error bars for corresponding values

https://preview.redd.it/ril37fptlco51.png?width=1650&format=png&auto=webp&s=d59bcfa739f75e97afaaa8fa765fb667a2384af4