Yes, the Lindeberg-Feller CLT works fine for (i).
Does anyone have any idea for case (ii)? That is, is there any CLT for the case the bernoulli trials are dependent with different success probabilities.
A Binomial distribution has a standard normal limiting distribution, i.e. (X-E[X])/se(X) -> N(0,1), where X is the sum of independent and identically distributed Bernoulli variables.
Does this hold even when
i) the Bernoulli variables are independent but non-identically distributed? That...
Let X1 and Y1 be two random variables. We have Cov(X1,Y1) = 0. Does this extend to any transformation X2 = g(X1) and Y2 = g(Y1), such that Cov(X2,Y2)? Here, g is a continuous function. For example, if we set X2 = X1^2 and Y2 = Y1^2. Do we then from Cov(X1,Y1) = 0 that Cov(X1^2,Y1^2) = 0?
Let x(1),...,x(N) all be independent uniformally distributed variables defined on (0,1), i.e. (x(1),...,x(N)) - U(0,1). Define the random variable y(i) = x(i)/(x(1)+...+x(N)) for all i=1,...,N. I’m looking for the pdf of the random variables y(1),…,y(N). Has anyone come across such random...
Let's say we have a cumulative distribution function (cdf) G and random numbers v1 and v2.
The definition of strict increasing function is: v1 < v2 => G(v1) < G(v2).
In a statistics book, the author writes:
"...but with the additional assumption that the cdf G is a strictly increasing...
I have two sets of deterministic numbers, collected in the two vectors: x=[x(1),...,x(n)] and y=[y(1),...,y(n)]. My (determinstic) theory says that x(i)=y(i) for all i=1,...,n. But instead, I want to assume that the numbers x and y are stochastic. If we let f(.) be a pdf, does this mean that I...