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Math Help - Please help: central limit example

  1. #1
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    Please help: central limit example

    I've been asked by a friend to help with a probability assignment... I don't know how to solve it so I figured I would post it here... thanks for the help!

    A fair die is tossed 40 times. Use the given equation (in the attachment) to determine the probability of the total number of dots being between 130 and 150.

    Repeat using the central limit theorem.

    thanks in advance!
    Attached Thumbnails Attached Thumbnails Please help: central limit example-imag0003.jpg  
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  2. #2
    MHF Contributor matheagle's Avatar
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    1 The CLT is more accurate than the attachment (Cheyshev's)
    2 Between 130, 150 is vague. You need to know if you're including the endpoints.

    This is the sum of 40 i.i.d. random variables with mean 3.5.
    As I guessed the mean of the sum is 40 times 3.5 which is.....140.
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  3. #3
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    okay sorry, i typed it word for word but i see now that it is vague. let us assume that 130 and 150 are included. what would the probability be using the given equation (which he is required to use for the first part)

    and then, using Central limit theorem, find the probability...


    thanks again!!!
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  4. #4
    MHF Contributor matheagle's Avatar
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    \sum_{i=1}^{40}X_i\approx N(\mu ,\sigma^2) via the CLT.

    Where \mu is the mean of the sum =(40)(3.5)=140.

    Likewise \sigma^2 is the variance of the sum, which is 40 times the variance of one X.

    So compute V(X) for a random variable with 1/6 probability on 1,2,3,4,5,6.
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  5. #5
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    I just saw the rest of your post... which answers most of my questions... i'll post in a minute with another question... i'm sure of it haha
    i've attached what the given textbook has for the CLT...
    Attached Thumbnails Attached Thumbnails Please help: central limit example-imag0005.jpg  
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    Var(X)=E[X^2]-E[X]^2 .... right? or is this a different equation
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  7. #7
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by mokash34 View Post
    Var(X)=E[X^2]-E[X]^2 .... right? or is this a different equation

    and M_n is just the sample mean, \bar X

    I thought M_n was the sum of the 40 random variables.

    It's the sum divided by 40.
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  8. #8
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    hmm... i think i'm in over my head... i don't seem to understand.

    this is one of three problems and i though it looked the easiest...

    so there will be 2 answers... one from CLT and one from the equation in my first post...

    is the CLT the equation you gave me? or is it the second picture i posted?

    are the answers going to be different?

    i understand if you feel like this is a waste of your time, thanks nonetheless
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  9. #9
    MHF Contributor matheagle's Avatar
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    Did you calculate the variance of one of these tosses?
    The CLT is much more accurate than chebyshev's.
    That's the point of this.
    Now it's not a binomial, in that case we add and subtract .5.
    We are approximating a discrete rv (number of spots) with a continuous rv (normal)
    so we should throw in some correction factor, but I doubt people will do that.
    That's why you should know if you're including the endpoints.
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  10. #10
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by mokash34 View Post
    Var(X)=E[X^2]-E[X]^2 .... right? or is this a different equation

    DO this for the distribution above.
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