Random variables
Random variables with zero correlation are not necessarily independent. Give a simple example.
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Let X be a normally-distributed random variable with
Mean zero. Let Y = X^2. Obviously, X and Y are not independent: knowing X, gives the value of Y.
The covariance of X and Y is Cov(X,Y) = E(XY) - E(X)E(Y) = E(X^3) - 0*E(Y) = E(X^3) = 0,
because the distribution of X is symmetric around zero. correlation r(X,Y) = Cov(X,Y)/Sqrt[Var(X)Var(Y)] = 0, the random variables are not independent, but correlation is zero.
Explain sampling bias and describe how random sampling serves to avoid bias in the process of data collection.
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In testing the null hypothesis H0: P=0.6 vs the alternative H1 : P < 0.6 for a binomial model b(n,p), the rejection region of a test has the structure X ≤ c, where X is the number of successes in n trials. For each of the following tests, d
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