Probability problems

1.22) An oil executive has determined that the probability that this oil field contains oil is 0.6. Before starting the drilling she decides to order a seismological test to see if there is oil in the ground. Unfortunately, the test is not entirely accurate. It concludes that there is oil with probability 0.9 if there is indeed ...continues

Probability

2.16) Consider the k-out-of-n system (info on the system: A system consists of n independent components. Each component functions with probability p. The system as a whole functions if at least k components are functioning (1 <= k <= n)). The probability that a component is functioning at time t is given to be e^(-t). Comput ...continues

Probability

2.20) Suppose a shock causes a unit damage to the machine with probability 0.1 and no damage with probability of 0.9. Successive shocks are independent. Suppose the damages are cumulative and the machine can withstand at most four units of damage. (That, is the machine fails when the fifth unit of damage is inflicted on it.) Wh ...continues

Probability

2.33) Suppose there are two vendors and each provides 50% of the items. The lifetime(in days) of an item from the first vendor is Exp(.1), and that from the second vendor is Exp(.08). Compute the probability that a randomly picked item will last more than 12 days.

Probability

2.54) Consider the k-out-of-n system (explanation: a system consists of n independent components. Each component functions with probability p. The system as a whole functions if at least k components are functioning (1 <= k <= n)). Suppose we visit this system at time t=3 and replace all failed components, at a cost of $75 eac ...continues

Probability

3.21) Suppose a machine has three independent components with Exp(.1) lifetimes. Compute the expected lifetime of the machine if it needs all three components to function properly. (This is a multivariate random variable problem)

Probability

4.19) The lifetimes of two car batteries (Brand A and B) are independent exponential random variables with means 12 hours and 10 hours, respectively. What is the probability that Brand B battery outlasts Brand A battery?

matrix equations

A fertiliser company has available 80 tonnes of nitrate and 50 tonnes of phosphate to use during the coming week to produce three different types of lawn fertilisers. The mixture ratios and profit are given in the table below . The manager of the company wants to maximise profit from the raw materials currently available and dec ...continues

Prove the inequality when n = 1, n = 2,

Please see the attached file for full problem description. Here is the problem Given and show that for some if then Hint: prove the inequality when n = 1, n = 2, and then do induction on n using the identity: of course, you have to proven the identity first.

Consider y = the least upper bound of...

Please see the attached file for full problem description. ALL STEPS MUST BE SHOWN PLEASE PROVIDE AN EXPLANATION IN COMPLETE SENTENCES Here is the problem Given and prove that there is a unique y>0 such that That is, exists and is unique Hint: Consider y = the least upper bound of Then use exercise 2 to s ...continues