+ Reply to Thread
Results 1 to 4 of 4

Thread: Central Limit Theorem

  1. #1
    Points: 3,621, Level: 37
    Level completed: 81%, Points required for next Level: 29
    askazy's Avatar
    Location
    Holand
    Posts
    160
    Thanks
    4
    Thanked 0 Times in 0 Posts

    Central Limit Theorem




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

  2. #2
    Points: 3,621, Level: 37
    Level completed: 81%, Points required for next Level: 29
    askazy's Avatar
    Location
    Holand
    Posts
    160
    Thanks
    4
    Thanked 0 Times in 0 Posts

    Re: Central Limit Theorem

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

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

  3. #3
    Points: 3,621, Level: 37
    Level completed: 81%, Points required for next Level: 29
    askazy's Avatar
    Location
    Holand
    Posts
    160
    Thanks
    4
    Thanked 0 Times in 0 Posts

    Re: Central Limit Theorem

    Could someone give me a light?

  4. #4
    Points: 3,621, Level: 37
    Level completed: 81%, Points required for next Level: 29
    askazy's Avatar
    Location
    Holand
    Posts
    160
    Thanks
    4
    Thanked 0 Times in 0 Posts

    Re: Central Limit Theorem


    Someone can take a look?
    The difference between stupidity and genius is that genius has its limits.
    "Albert Einstein"

+ Reply to Thread

           




Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts






Advertise on Talk Stats