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Probability and Sampling Distributions

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  1. Probability and Sampling Distributions http://mikeess-trip.blogspot.com/2011/06/gambling.html

  2. Frequentist Interpretation

  3. Sample Space What is the sample space in the following situations: • I roll one 4-sided die. • I roll two 4-sided dice. • I toss a coin until the first head appears. • A mortgage can be classified as fixed rate (F) or variable (V) and we are considering 2 houses. • The number of minutes that a college student uses their cell phone in a day.

  4. Examples: Probability Rules Compliment: • Roll two 4-sided dice: What is the probability that the sum is greater than 3? • Mortgage: What is the probability that both houses do not have fixed mortgages? Additivity: • Roll two 4-sided dice: What is the probability that the sum is 2 or 3? • Mortgage: What is the probability that both houses have the same type of mortgage?

  5. Example: Types of Probabilities For each of the following, determine the type of probability and then answer the question. • What is the probability of rolling a 2 on a fair 4-sided die? • What is the probability of having a girl in the following community? • What is the probability that the US will win the most medals in the Winter Olympics?

  6. Examples: Probability Histograms

  7. Example: Probability A person casually walks to the bus stop when the bus comes every 30 minutes has a density function of What is the probability that a person will have to wait between 5 and 10 minutes?

  8. Normal Distribution: Example A particular rash has shown up in an elementary school. It has been determined that the length of time that the rash will last is normally distributed with mean 6 days and standard deviation 1.5 days. • What is the percentage of students that have the rash for longer than 8 days? • What is the percentage of students that the rash will last between 3.7 and 8 days?

  9. Example: Expected value What is the expected value of the following a) A fair 4-sided die b)

  10. Example: Expected Value An individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is a) HW: Verify that E(X) = 0.60. b) If the cost of insurance depends on the following function of accidents, g(y) = 400 + (100y -15), what is the expected value of the cost of the insurance?

  11. Example: Expected Value 5 individuals who have automobile insurance from a certain company are randomly selected. Let X and Y be two different accident profiles in this insurance company: E(X) = 0.60 E(Y) = 0.95 a) What is the expected value the total number of accidents of the people if 2 of them have the distribution in X and 3 have the distribution in Y?

  12. Example: Expected value If the density function is Calculate E(X2).

  13. Example: Variance If the density function is Calculate Var(X).

  14. Correlation

  15. Example: Variance An individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is a) HW: Verify that Var(X) = 0.74. b) If the cost of insurance depends on the following function of accidents, g(y) = 400 + (100y -15), what is the standard deviation of the cost of the insurance?

  16. Example: Variance 5 individuals who have automobile insurance from a certain company are randomly selected. Let X and Y be two different accident profiles in this insurance company: Var(X) = 0.74 Var(Y) = 0.95 a) What is the standard deviation of the difference between the 2 who have insurance using the X distribution and the 3 who have insurance using the Y distribution?

  17. Spread as a function of n

  18. Example: mean and SD of Sampling Distribution The time that it takes a randomly selected rat of a certain subspecies to find its way through a maze has a normal distribution with μ = 1.5 min and σ = 0.35 min. Suppose five rats are random selected. • What is the mean of the average time? • What is the standard deviation of the average time?

  19. Example – Sampling Distribution: Normal The time that it takes a randomly selected rat of a certain subspecies to find its way through a maze has a normal distribution with μ = 1.5 min and σ = 0.35 min. Suppose five rats are random selected. • What is the probability that the average time is at most 2.0 minutes? • What is the probability that the average time will be within 0.3 minutes of the mean?

  20. CLT: Example 1 (in class) An electronics company manufactures resistors that have a mean resistance of 100 ohms and a standard deviation of 10 ohms. Assume that the distribution of resistance is normal. a) Find the probability that one resistor will have a resistance less than 95 ohms. (0.3085) b) Find the probability that a random sample of 25 resistors will have an average resistance less than 95 ohms. (0.0062)

  21. CLT: Example 2 (in class) Without checking the city bus web site, a student walks at random times to the Beering Hall bus stop to wait for the Ross Ade bus which is supposed to arrive every 10 minutes. This will be a Uniform distribution with 0 ≤ x ≤ 10. For a Uniform distribution on the interval (a,b), a) If one student walks to the bus stop to catch this bus, what is the probability that the wait time will be more than 6 minutes? (0.4) b) If 40 students walk to the bus stop to catch this bus, what is the probability that the average wait time will be more than 6 minutes? (0.0143)