### Learning Goals

To understand the Central Limit Theorem and be comfortable using it to approximate otherwise computationally demanding statistics.

None!

### Concept Check

Q: Can we write iid rather than i.i.d. in our work?

Q: Is Exp(0) allowed?

Q: Does the error tend to 0 or the error divided by n or soemthing else?

Q: so the central limit theorem is an approximation?

A1:  yes!

Q: I can’t see any of the questions being asked for some reason.

A1:  I couldn’t either. But I just was able to… not sure what zoom is up to

Q: so iid and CLT is only useful for sampling? since they need to have the same distribution while independent of each other.

A1:  I would say, rather, that all samples are assumed to be IID, but not all IID variables must be samples. CLT is slightly more general. Any sum of random variables which are independent and identically distribution is normal…

Q: so by max, we mean the probability that the max of all the random variables is a certain value? For example, what is the probability the max of 100 die rolls is a 5?

A1:  exactly! super interesting. Shows up a lot…

Q: If we assume the Central Limit Theorem, does this imply that the sum of two gaussians must be a gaussian as the gaussians are essentially a fixed point?

A1:  thats a smooth interpretation. yes :)

Q: the continuity correction is because we are using a cont distribution to approximate a discrete one?

A1:  exactly! Sum of dice is discrete… gaussian is cont.

Q: ^^ I didn’t understand why CLT proves the sum of gaussians is a gaussian – doesn’t CLT require that the independent variables are identically distributed?

A1:  actually good question. I can only use it to prove sum of gaussians if they come from the sum of the same underlying IID variables…

Q: Can you explain why it's 100/n again?

A1:  That was a trick to express your variable as a sum.. so you could use CLT… long story short, poissons with large lambda can be approximated by a normal

Q: when will the lecture on the beta distribution be?

A1:  March 1st!