I'm not sure how F_Z(1), F_Z(2), … , F_Z(N) the distribution functions, which may or may not be identical, were employed to come closer to F_Z(z). If I had to guess, it's possible that the parameter(s) for F_Z(i), were approximated in each trial, bringing closer and closer to F_Z(z) . In other examples, some use a single distribution function i.e. F_N(X_N) to converge F(X). Please let me know whether this viewpoint meets the criteria for convergence.
The basic idea he assumed (or maybe missed) as implicit is the whole key to understanding what it means to converge in distribution as n tends to infinity. Basically, many times ( not always), when you write a r.v. as Z_n, you say that the distribution of Z_n *depends on n* (VERY IMPORTANT) For example let's consider a sequence of r.v. X_1, X_2, X_3....X_n where X_i ~ G_i, which may or may not depend on n. and define Z_i = h(X_1, X_2,...X_n, n) Then each Z_i ~ F_i, which depends on n. As n tends to infinity, Z_n converges in distribution to Z, where Z ~ F(z). If this sounds complicated, reply to my comment and I will write this in simplified language with an example.
Thank you very much, this really saves my life!
You explaining this thing in 7 minutes is much clearer than my professor doing it half a semester. Thank you!
Your professor took half a semester to explain convergence in distribution? That's worrisome.
Thank you sir. It's helpful
perfecto!
Thank you sir
Is this not in the playlist yet
I'm not sure how F_Z(1), F_Z(2), … , F_Z(N) the distribution functions, which may or may not be identical, were employed to come closer to F_Z(z).
If I had to guess, it's possible that the parameter(s) for F_Z(i), were approximated in each trial, bringing closer and closer to F_Z(z) .
In other examples, some use a single distribution function i.e. F_N(X_N) to converge F(X).
Please let me know whether this viewpoint meets the criteria for convergence.
Are Z1, Z2, . . . , Zn themselves random variables representing the average of 1, 2, . . . , n samples (respectively) drawn from a population?
Z1, Z2, . . . , Zn are random variables, not related to samples or realization
thank you
thx!! but, still can't understand this. could you give a more detailed example? such as giving the pdf of some distribution...
The basic idea he assumed (or maybe missed) as implicit is the whole key to understanding what it means to converge in distribution as n tends to infinity.
Basically, many times ( not always), when you write a r.v. as Z_n, you say that the distribution of Z_n *depends on n* (VERY IMPORTANT)
For example let's consider a sequence of r.v. X_1, X_2, X_3....X_n where X_i ~ G_i, which may or may not depend on n.
and define Z_i = h(X_1, X_2,...X_n, n)
Then each Z_i ~ F_i, which depends on n.
As n tends to infinity, Z_n converges in distribution to Z, where Z ~ F(z).
If this sounds complicated, reply to my comment and I will write this in simplified language with an example.
@@abhishekbhatia6092 can you simplify this
@@abhishekbhatia6092 Yah, simplification would be pleasing :-).
Was hoping this video can explain the continuity part, but he said he will skip it. Oh well.
Thank you sir