If we have a random sample X_1,...,X_n chosen from an infinite population or a finite population {s_1,...,s_N} with replacement, and each individual in the population has a pdf f(x), then we can say the X_i are iid and the joint pdf over the sample is
f(x_1,...,x_n)=f(x_1)f(x_2)***f(x_n)
First, am I correct to say that if the population is finite, then the $x_i\in\{s_1,...,s_N\}$?
Now, I am trying to get a grasp of the situation of sampling from a finite population without replacement in comparison to the preceding situation. My book (Statistical Inference by Casella and Berger) assumes that the sampling is done uniformly over {s_1,...,s_N}, and so while the marginal distribution of X_1 is P(X_1=x)=1/N, the marginal distribution of X_2 is
P(X_2=x)=sum_{i=1}^N P(X_2=x|X_1=x_i)P(X_1=x_i)
which eventually simplifies to 1/N as well. How does this compare to the situation of sampling with replacement? Has f(x) been declared as uniform, or is the uniform distribution referring to something else? Can we say the marginal distributions of the X_i are iid, and if so does that mean there is some other underlying distribution across the population which is conditionally altered in the sampling? Also does that mean given this underlying distribution f(x), can we write the joint distribution the same way as in the with replacement case above? If not, what would the joint pdf look like? Lastly, I've been saying pdf but does sampling without replacement imply a discrete distribution, or am I getting confused between the discreteness of the population and discreteness of the underlying distribution (and if so, how? I originally thought the s_i were just different objects in the population from which observations x_i are made and assigned as realizations to the X_i, but I'm not too sure of that anymore since I don't know how that fits into the model described).
This may sound dumb but I’m just starting out too and it’s helped me a lot. Literally just recreate your favorite shit. Take your favorite song and remake the beat from the ground up to the best of your ability. Doing this will help you understand the mindset of the producer and what he’s listening for when he chops up those samples.
Same thing for verses: take a verse from one of your favorite artists, rip off the entire flow and fill in your own lyrics. This will hopefully help you figure out what great rappers are listening for in their beats and how they really dive into the crevices of the music. I know it feels lame as fuck, but this is not stuff anyone will ever hear, it’s simply for your own advancement. Also: you are lame as fuck, and will be for a while until you get the hang of the whole writing thing.
I’d love to hear other people’s thoughts on this, it’s something that’s really helped me personally
I have a TON of samples I've made, and am constantly making them and not (really) using them i make samples in Many different genres such as <edm, chiptune, hip hop, red, but I've mostly just been focusing on my sound. I'm basically asking if anyone would use samples that are in my style of music more than any of those
Tl:DR: If I post samples I've made will you use them?
The title is a quote from the OpenIntro Statistics textbook. The population is college students with the values being how many hours of sleep they get. I do not understand why the size of the sample, in this case less than 10%, will determine if the sample is independent or not?
