1. Determine the expected value for each of the following.
(a) f(x)= (1/√x)k for 1 ≤ x ≤ 4
(b) f(x)= k ln x for 1 ≤ x ≤ 2
(c) f(x) = ke-x for 0 ≤ x ≤1
(d) f(x) kex for 0 ≤ x ≤ 1
2. Determine the variance for each of the following functions.
(a) f(x)= (1/√x)k for 1 ≤ x ≤ 4
(b) f(x)= k ln x for 1 ≤ x ≤ 2
(c) f(x) = ke-x for 0 ≤ x ≤1
(d) f(x) kex for 0 ≤ x ≤ 1
3. The probability of a failure of a certain part is described by the density function f(t) = kt2 for 0 ≤ t ≤ 10. Determine the probability of failure for t ≤ 5 and find the expected value and variance of the time to failure.
4. The time between telephone calls at a resort hotel is distributed according to the exponential distribution with an expected time between calls of 4 minutes. Determine the probability of a call within 2 minutes.
5. The probability of failure of a battery is described by the probability density function f(t) = kt2 for 0 ≤ t ≤ 120 hours. Determine the probability of the batteries failing within 60 hours.
6. The marginal cost of manufacturing an additional unit is known to be related to the total number of units manufactured. If this relationship can be described by the function f(x) = 100 - 4x (1/100)x2, determine the total cost function. Assume that the fixed cost is $500.00.