. Suppose Z has a standard normal distribution. (a) Find P(−1 ≤ Z ≤ 1), P(−1 ≤ Z ≤ 2), P(−2 ≤ Z ≤ −1), P(−∞ ≤ Z ≤ 1). (b) If P(Z ≤ a) = 0.45 and P(0 ≤ Z ≤ b) = 0.45, find a and b. [Hint: Use qnorm in...

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Suppose Z has a standard normal distribution.

(a) Find P(−1 ≤ Z ≤ 1), P(−1 ≤ Z ≤ 2), P(−2 ≤ Z ≤ −1), P(−∞ ≤ Z ≤ 1).

(b) If P(Z ≤ a) = 0.45 and P(0 ≤ Z ≤ b) = 0.45, find a and b. [Hint: Use

qnorm in R.]

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In a certain genome the bases appear to be iid and pG = 0.3.

Define the (binomial) count of the number of Gs in the first 1000 bases as

N = X1 + X2 + · · · + X1000.

(a) Give the mean and variance of N.

(b) Approximate, using the Central Limit Theorem, P(0 ≤ N ≤ 329) and

P(285.5 ≤ N ≤ 329).

(c) Produce a histogram for 1000 replicates of N and compare the results

with those of (b).