b. ( ) ( ) ( ) ( ) ( ) 5. Independence: Two events (A & B) are independent if one of the conditions listed below is satisfied; ( ) ( ) ( )
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1 1. Set a. b. 2. Definitions a. Random Experiment: An experiment that can result in different outcomes, even though it is performed under the same conditions and in the same manner. b. Sample Space: This is the set of all possible outcomes. i. Discrete: Consists of finite or countable infinite set of outcomes ii. Continuous: Contains an interval (either finite or infinite of real numbers c. Event: It is a subset of the sample space of a random experiment. i. Mutually Exclusive: 3. Probability a. b. c. d. 4. Conditional Probability: a. b. 5. Independence: Two events (A & B are independent if one of the conditions listed below is satisfied; a. b. c. 6. CIRCUIT PROBLEM:- A 0.99 B 0.99 C 0.8 Fig 2:- Shows the probability of Success ( ( 7. a. b. c. d. 8. Mean (Expected Value: 9. Variance: 10. Standard Deviation: 11. Discrete Uniform Distribution (DUD: This is when every member of the sample space has the same probability. a. For DUD, the b. For DUD, the 12. Bernoulli Trials:- A random experiment consists of Bernoulli trials if; - The trials are independent. - Each trial results in only two possible outcomes (Success and Failure. - The probability of success in each trial remains constant a. Binomial Random Variable:- A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents an independent trial. a Over five mornings, what is the probability that the light is green on exactly one day? Page 1 of 6
2 The probability of a successful optical alignment in the b. Geometric Random Variable:- assembly of an optical data storage product is 0.8. Assume the trials are independent. a What is the probability that the first successful alignment requires exactly four trials? c. Negative Binomial:- 13. Hypergeometric: - A state runs a lottery in which six numbers are randomly selected from 40, without replacement. A player chooses six numbers before the state s sample is selected. a What is the probability that the six numbers chosen by a player match all six numbers in the state s sample? 14. Poisson Distribution:- This refers to any random experiment with the properties below; Counting the number of occurrence of an event over a period of time/space. - The # of occurrence of an event in an interval is proportional to the length of the interval. - Events cannot occur simultaneously. - Occurrences of the events are independent for non-overlapping intervals. 15. Continuous Random Variable (C.R.V 16. Continuous Uniform Distribution: a. For CUD, the b. For CUD, the ( 17. Normal Distribution: Note: you can only read from the table when standardized a. 18. Normal Approximation to Binomial Distribution: Use iff both a. b. ( c. ( Page 2 of 6
3 19. Normal Approximation to Poison RV: Use iff both a. 20. Exponential Distribution: Amount of wait time until first count is obtained use integral. ( a. Lack of Memory Property: What is the probability that EQ is detected before 2 years if it doesn t occur the first year? EQ occurs at a rate of 1.5 yearly. 21. a. b In a data communication system, several messages that arrive at a node are bundled into a packet before they are transmitted over the network. Assume the messages arrive at the node according to a Poisson process with messages per minute. Five messages are used to form a packet. a What is the mean time until a packet is formed, that is, until five messages have arrived at the node? b What is the probability that a packet is formed in less than 10 seconds? 22. Weibull Dist. : [ [ [ Suppose the lifetime of a component (in hours is modeled with a Weibull distribution with and. Determine the following: (a (b (c value for x such that ( ( ( ( ( ( 23. Lognormal Dist. : [ [ ( ( [ 24. Beta Dist. : ( ( Suppose that X has a lognormal distribution with parameters and. Determine the following: (a (b (c Suppose X has a beta distribution with parameters and. Determine the following: (a (b (c mean and variance ( [ [ [ [ ( ( [ [ Page 3 of 6
4 25. Joint Probability: [ [ [ [( [( [ [ 1. Marginal Probability: 2. Conditional Probability: [ [ [ [ ( ( Covariance : Correlation : Page 4 of 6
5 Page 5 of 6
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