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Suppose that Xi are IID normal random variables with mean 5 and variance 10, for i...

Suppose that Xi are IID normal random variables with mean 5 and variance 10, for i = 1, 2, ... , n.

(a) Calculate P(X1 < 6.2), i.e., the probability that the first value collected is less than 6.2.

(b) Suppose we collect a sample of size 2, X1 and X2. What is the probability that their sample mean is greater than 6.3?  

(c) Again, suppose we collect two samples (n=2), X1 and X2. What is the probability that their sum is greater than 11.4? (Hint: P(X1 + X2 > 11.4) = P( X > 5.7).

(d) Suppose we collect a sample size of n=100. What is the probability that the sample mean is between 4.53 and 5.32?

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