(a) Consider the function f(x) = 1 4 e − x 4 , x ≥ 0; 0, otherwise. (i) What properties must f have in order to be a val

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(a) Consider the function f(x) = 1 4 e − x 4 , x ≥ 0; 0,
otherwise.
(i) What properties must f have in order to be a valid
probability density function?
(ii) Calculate the expected value (mean) and variance.
(iii) Find P(X ≥ 4).
(b) Items from a production line can be classified as either
meeting the customer specifications or not meeting customer
specifications. The probability that an item meets customer
specification is 0.80. Find the probability that exactly 10 out of
the next 15 items produced by the line meet customer
specifications.
(c) Mathematics Scholastic Aptitude Test (SAT) scores are
approximately normally distributed with mean 500 and standard
deviation 100, so if X represents a mathematics SAT score, X ∼
N(500, 100). What is the probability that an individual mathematics
SAT score is 550 or more?