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Math Help - Independent Random Variables

  1. #1
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    Independent Random Variables

    3. Suppose we roll a dice n times. Define the random variable Xi to be equal to 1 if the
    ith roll results in an even number, and 0 otherwise. Define the the random variable Yi to be
    equal to 1 if the ith roll results in the number 6, and zero otherwise.
    (a) Are X1 and Y1 independent random variables?
    (b) Find the distribution of Z1 = X1 + Y1.
    (c) Suppose n is 500. What is the approximate distribution of
    1/500 (500)sigma(i=1)Xi +1/500 (500)sigma(i=1)Yi
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  2. #2
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    Quote Originally Posted by farshyfaroo View Post
    3. Suppose we roll a dice n times. Define the random variable Xi to be equal to 1 if the
    ith roll results in an even number, and 0 otherwise. Define the the random variable Yi to be
    equal to 1 if the ith roll results in the number 6, and zero otherwise.
    (a) Are X1 and Y1 independent random variables?
    X_1 and Y_1 are independednt if:

    P( (X_1=a) \wedge (Y_1=b))=P(X_1=a)P(Y_1=b)

    for all a and b in the appropriate sample spaces. Let a=b=1, does this hold?

    (b) Find the distribution of Z1 = X1 + Y1.
    The possible values of Z_1 are 0, 1, 2 calculate their probabilities.

    (c) Suppose n is 500. What is the approximate distribution of
    1/500 (500)sigma(i=1)Xi +1/500 (500)sigma(i=1)Yi
    U=\frac{1}{500}\sum_{i=1}^{500} X_i+\frac{1}{500}\sum_{i=1}^{500} Y_i=\frac{1}{500}\sum_{i=1}^{500} Z_i

    Now the Z_i 's are iid so the U is the mean of a sample of size 500 for a sample from the same distribution as Z_1 ... so is approximatly normally distributed with mean and variance ...

    CB
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