Central Limit Theorem

askazy

New Member
#1
If X is a Gamma(n,1). For which values of n we have
[math]P(|\frac{X}{n}-1|>0.01)<0.1[/math]
If [math]n=n_0[/math] is the first value of n that satisfies the above equation, so find the n values for
i) [math]n=n_0[/math]
ii) [math]n=n_0+1[/math]
iii)[math]n=n_0+2[/math]
Use chi-square distribution or Central Limite Theorem
 

askazy

New Member
#2
What I think
Supose that [math]X_i = exp(1) \rightarrow \sum_{i=1}^nX_i \rightarrow Gamma(n,1)=X[/math] now
[math]P(\frac{X}{n}-1>0.01) = 1 - P(X-n<0.01n)=1-P(X<1.01n)[/math]
We have [math]E[X_i]=1\rightarrow\sum_{i=1}^nE[X_i]=n[/math] analogously [math]Var(X_i)=1\rightarrow\sum_{i=1}^nVar(X_i)=n[/math]
applying Central Limit Theorem
[math]1 - P(\frac{\sum_{i=1}^nX_i - \sum_{i=1}^n\mu_i}{\sqrt{\sum_{i=1}^n\sigma_i^2}}\leq\frac{1.01n-n}{\sqrt{n}})=1-P(Z\leq0.01\sqrt{n})<0.1[/math][math]=-P(Z\leq0.01\sqrt{n})<-0.9[/math]*

I do not know if I applied the CLT correctly, and the last inequality marked with * do not remember how to solve it.