Copyright ©2011 Brooks/Cole, Cengage Learning Continuous Random Variables Class 36 1

Homework Check Assignment: Chapter 8 – Exercise 8.25, 8.27 and 8.29 Reading: Chapter 8 – p

Suggested Answer Copyright ©2011 Brooks/Cole, Cengage Learning 3

Suggested Answer Copyright ©2011 Brooks/Cole, Cengage Learning 4

5 Stability or Excitement Two plans for investing $100 – which would you choose? What is the Expected Value for each plan:

Copyright ©2011 Brooks/Cole, Cengage Learning 6 Example 8.13 Stability or Excitement Two plans for investing $100 – which would you choose? Expected Value for each plan: Plan 1: E(X ) = $5,000  (.001) + $1,000  (.005) + $0  (.994) = $10.00 Plan 2: E(Y ) = $20  (.3) + $10  (.2) + $4  (.5) = $10.00

Copyright ©2011 Brooks/Cole, Cengage Learning 7 Example 8.13 Stability or Excitement Variability for each plan: Plan 1:V(X ) = $29,900.00and  = $ Plan 2: V(X ) = $48.00and  = $6.93 The possible outcomes for Plan 1 are much more variable. If you wanted to invest cautiously, you would choose Plan 2, but if you wanted to have the chance to gain a large amount of money, you would choose Plan 1.

From Discrete to Continuous Random Variables Copyright ©2011 Brooks/Cole, Cengage Learning 8

9 8.5 Continuous Random Variables Continuous random variable: the outcome can be any value in an interval or collection of intervals. Probability density function for a continuous random variable X is a curve such that the area under the curve over an interval equals the probability that X is in that interval. P(a  X  b) =area under density curve over the interval between the values a and b.

Discrete Variable VS Continuous Variable Discrete Variable –Roll a die – outcomes are a set of isolated (“Discrete”) values. –Therefore, we can determine the probability of a single value occurring (Example, P (4) =1/6) Continuous Variable –The travelling speed of a car – outcomes are a set of continuous values –The probability of any specific outcome is always zero because the total possible outcomes is infinitive. –Therefore, we interest to find the probability that an event falls within a range 10

Copyright ©2011 Brooks/Cole, Cengage Learning Normal Random Variables Most commonly encountered type of continuous random variable is the normal random variable, which has a specific form of a bell-shaped probability density curve called a normal curve. A normal random variable is also said to have a normal distribution Any normal random variable can be completely characterized by its mean, , and standard deviation, .

Normal Probability Distribution Curve Standard Normal Distribution Curve 1.Symmetric, Unimodal and bell shape 2.Areas under a standard normal distribution curve represent proportions of the total number of observations 3.The area under the curve = 1 or 100% 12

Copyright ©2011 Brooks/Cole, Cengage Learning 13 Example 8.21 College Women’s Heights Data suggest the distribution of heights of college women described well by a normal curve with mean  = 65 inches and standard deviation  = 2.7 inches. Draw a normal curve to represent the above data

14 Example 8.21 College Women’s Heights – Check your graph Note: Tick marks given at the mean and at 1, 2, 3 standard deviations above and below the mean. Now incorporate the Empirical Rule of a Normal Curve in your drawing

Copyright ©2011 Brooks/Cole, Cengage Learning 15 Example 8.21 College Women’s Heights – Check your graph Note: Tick marks given at the mean and at 1, 2, 3 standard deviations above and below the mean. Empirical Rule are exact characteristics of a normal curve model.

Copyright ©2011 Brooks/Cole, Cengage Learning 16 Useful Probability Relationships

Copyright ©2011 Brooks/Cole, Cengage Learning 17 Useful Probability Relationships

Copyright ©2011 Brooks/Cole, Cengage Learning 18 Example 8.22 Prob for Math SAT Scores Math SAT scores have a normal distribution with mean  = 515 and standard deviation  = 100. Q: Probability a score is less than or equal to 600?

Copyright ©2011 Brooks/Cole, Cengage Learning 19 Example 8.22 Prob for Math SAT Scores Check your answer Q: Probability a score is less than or equal to 600? Answer :.8023.

Copyright ©2011 Brooks/Cole, Cengage Learning 20 Example 8.22 Prob for Math SAT Scores Math SAT scores have a normal distribution with mean  = 515 and standard deviation  = 100. Q: Probability a score is greater than 600?

Copyright ©2011 Brooks/Cole, Cengage Learning 21 Example 8.22 Prob for Math SAT Scores check your answer Q: Probability a score is greater than 600? Answer: By complement, 1 –.8023 =.1977

Copyright ©2011 Brooks/Cole, Cengage Learning 22 Example 8.22 Prob for Math SAT Scores Math SAT scores have a normal distribution with mean  = 515 and standard deviation  = 100. Q: Probability a score is between 515 and 600?

Copyright ©2011 Brooks/Cole, Cengage Learning 23 Example 8.22 Prob for Math SAT Scores – Check your answer Answer : (since 50% of scores are larger than the mean of 515),.8023 –.5000 =.3023

Copyright ©2011 Brooks/Cole, Cengage Learning 24 Example 8.22 Prob for Math SAT Scores Math SAT scores have a normal distribution with mean  = 515 and standard deviation  = 100. Q: Probability a score is more than 85 points away from the mean in either direction?

Copyright ©2011 Brooks/Cole, Cengage Learning 25 Example 8.22 Prob for Math SAT Scores check your answer Math SAT scores have a normal distribution with mean  = 515 and standard deviation  = 100. Q: Probability a score is more than 85 points away from the mean in either direction? A: (since 600 is 85 points above the mean) =.3954.

Calculate the probability of X that falls within a range Given: A check point is set up to check the speed of cars Assume the speeds are normally distributed. Mean is 61 miles per hour and the standard deviation is 4 miles per hour Calculate the probability that the next car that passes through the checkpoint will be traveling slower than 65 miles per hour. 26

Steps –Calculate the standard score for x = 65 Z-Score = (Observation – mean)/standard deviation –Use the Z-score table to find the percentile P(x<65) = P ( z< 1) = –Interpretation: The probability of the next car passing the check point and travelling slower than 65 miles per hour is

Quick Check 1 Based on the same info given in the sample, calculate the probability that the next car passing will be traveling more than 66 miles per hour. 28

Quick Check 2 Given: –A check point is set up to check the speed of cars –Assume the speeds are normally distributed. –Mean is 61 miles per hour and the standard deviation is 4 miles per hour a.Calculate the probability that the next car that passes through the checkpoint will be traveling less than 59 miles per hour. b.Calculate the probability that the next car that passes through the checkpoint will be traveling between 55 and 65 miles per hour 29

Homework Assignment: Chapter 8 – Exercise 8.63, 8.65, 8.67 and 8.69 Reading: Chapter 8 – p