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 p

[answerhappygod]

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? [2 marks] (ii) Calculate the expected value
(mean) and variance. [3 marks] (iii) Find P(X ≥ 4). [3 marks] (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. [6 marks] (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? [6 marks]