stats new work

here

2. Conditional probability and Bayes’ rule
A network intrusion software uses two systems to detect intruders. If an intruder is present, system A sounds the alarm 90% of the time, while system B sounds the alarm 95% of the time. If no intruder is present then system A sounds the alarm 20% of the time and system B only 10% of the time. You may assume that under a given condition (intruder present or not) the two systems operate independently
a) If there is an intruder, what is the probability that both systems sound an alarm?
b) If there is no intruder, what is the probability that both systems sound an alarm?
c) If there is an intruder, what is the probability that at least one of the alarms will sound?
d) A particular network has an intruder with 40% probability. Given that both alarms sound, what is the probability than an intruder is present in the network?

3. Discrete and continuous random variables
For budgeting purposes, each manager in a company must estimate their group’s monthly expenses. Manager 1 estimates that her monthly expenses are uniformly distributed between $15,000 and $45,000. Manager 2 estimates that his monthly expenses are normally distributed with mean $21,000 and standard deviation $3,000. Manager 3 estimates that her monthly expenses follow a discrete distribution with p($15, 000) = 0.2, p($20, 000) = 0.4, p($25, 000) = 0.3, and p($30, 000) = 0.1.
a) For each of the three managers, compute the mean and standard deviation of their group’s monthly expenses.
b) For each of the three managers, estimate the probabilities that their group’s monthly expenses are: i) between $17,000 and $24,000, ii) higher than $22,000, iii) below $18,000, and iv) exactly $25,000.
c) The company wants to budget enough for each group that the probability of expenses being over budget is at most 15%. How much should the company budget for each manager?
d) The company wants to budget enough for each group that the probability of expenses being over budget is at most 0.5%. How much should the company budget for each manager?
4. The binomial distribution
A manufacturer produces 1000 computer chips for a mission-critical application. Each chip costs $100 to manufacture and sells for $2000, but has a 1% chance of being defective.
a) Compute the mean and standard deviation of the number of defective chips.
b) Assume that 99.9% of defective chips are discovered (and destroyed) before they are sold, while the other 0.1% are sold with defects. The manufacturer suffers an estimated loss of $20,000,000 for each defec- tive chip sold (due to lawsuits, penalties, and bad public relations). What is the manufacturer’s expected profit?
c) The manufacturer develops a new quality control technology which enables detection of 100% of de- fective chips. How much would the implementation of this technology increase the manufacturer’s expected profit?