Chapter 2 - Review of statistical models

Question 1

discrete random variable

Answer 1

all possible values of X are countably infinite (or finite)

Question 2

probability mass function

Answer 2

sum p(x) = 1

Question 3

continuous random variables

Answer 3

range space is an interval (or collection of intervals) - get probability as integral from a to b

Question 4

cumulative distribution function

Answer 4

sum from lowest value in range to current value

Question 5

expectation

Answer 5

average or mean
- the expected value of X found by summing probability of X occuring by X

Question 6

n-th moment

Answer 6

expectation is the first moment of X - variance is the second

Question 7

variance

Answer 7

the expectation of (X - E(X))^2

Question 8

mode

Answer 8

discrete = value that occurs most frequently
continuous = where probability distribution function is maximized

Question 9

Queuing systems

Answer 9

• probabilistic models used to simulate queuing systems
• uses gamma, exponential, and Weibull distributions to model interarrival and service times

Question 10

Central Limit Theorum

Answer 10

• if have huge sample, distribution approaches a normal distribution
• may not always be appropriate
• all our tests rely on this theorum (ex. Z or T tests)

Question 11

Dominant characteristics of a probability distribution

Answer 11

• mean, variance, and mode
  (mean and mode are the same for a gaussian distribution)

Question 12

Inventory and Supply Chain

Answer 12

• has at least 3 random variables (number of units demanded per order/time period, time between demands, the lead time - time between placing order for stocking and receive order)
  -lead-time distribution often modelled by a gamma distribution
• geometric, Poisson, and negative binomial distributions are provide satisfying shapes

Question 13

Reliability and Maintainability

Answer 13

• exponential, gamma, and Weibull distribution can be used to model time to failure
• random failure = exponential
• standby redundancy (each component has an exponential time to failure) = gamma
• many components and failure due to serious number of defects = Weibull

Question 14

Bernoulli Trials/Distributions

Answer 14

• n trials that can be success or failure
• mean [E(X)] = p
• Variance [V(X)] = p(1-p)

Question 15

Binomial distribution

Answer 15

• number of successes in n Bernoulli trials
• E(X) = np
• V(X) = npq

Question 16

Geometric distribution

Answer 16

• probability that X will be the first success
• E(X) = 1/p
• V(X) = q/p^2

Question 17

Poisson

Answer 17

• when alpha > 0:
• E(X) = V(X)

Question 18

Uniform Distribution

Answer 18

• probability of each X is equal

Question 19

Poisson Process

Answer 19

• counting process - a stochastic process is a counting process if N(t) represents the total number of events taht occur by time t
• Assumes:
  1. arrivals occur one at a time
  2. stationary increments - distribution of the numbe rof arrivals between t and t+s depends on the interval s not on t (arrivals are completely random)
  3. independent increaments - number of arrivals during nonoverlapping time intervals are independent random varaibles
• Poisson arrival processes can be split into 2 independent Poisson arrival processes with probability p and 1-p adn arrival rates of lambda(p) and lambda(p-1)