Continuous Random Variables

1. Continuous Random Variables Week 6, Friday

2. Continuous Random Variable The Discrete Random Variables we’ve seen so far are easy to represent in table form; however, some random variables are impossible to represent in this form. Example: Consider the number of seconds it takes to run one mile. For very fast runners, this number could be around 300. For very slow runners, this number could be as high as 1,200. Furthermore, it is possible to get more precise values by rounding to some decimal, for example 400.25 seconds. To put all of these values into a table would be impractical. For the purpose of this class, we will define such random variables as continuous random variables. But if we don’t define X by a table then how do we define it? ANSWER: with a function

3. Examples of a Continuous R.V. Suppose I give a 10 point quiz. There are 100 questions, and partial credit is possible for each one. Then the score for students would be a continuous random variable because decimal scores such as 8.38495/10 may be possible, and these values are tough to put into a table. The way we describe such a random variable is through a function. Consider the following function as a probability model: where x takes on values between 0 and 10. What is the Area Under This Curve? 10 * (1/10) = 1.00 = 100% P[0 ≤ X ≤ 10] = 100% How does this link to Probability? For continuous random variables, P[a ≤ X ≤ b] can be found by calculating the area under the curve, between x=a and x=b 0 10

4. Examples of a Continuous R.V. Suppose I give a 10 point quiz. There are 100 questions, and partial credit is possible for each one. Then the score for students would be a continuous random variable because decimal scores such as 8.38495/10 may be possible, and these values are tough to put into a table. The way we describe such a random variable is through a function. Consider the following function as a probability model: where x takes on values between 0 and 10. What is the Area Under This Curve? P[passing] = ? 10 * (1/10) = 1.00 = 100% P[6 ≤ X ≤ 10] = 4/10 = 40% How does this link to Probability? For continuous random variables, P[a ≤ X ≤ b] can be found by calculating the area under the curve, between x=a and x=b 0 10 6

5. Examples of a Continuous R.V. Suppose I give a 10 point quiz. There are 100 questions, and partial credit is possible for each one. Then the score for students would be a continuous random variable because decimal scores such as 8.38495/10 may be possible, and these values are tough to put into a table. Alternative model: Define the function: What is the Area Under This Curve? 10 * (.2) * (1/2) = 1.00 = 100% Now, what is the probability of Passing the quiz? P[passing] = 1 – area of the small triangle = 1 – ( 4 * .2 / 2 ) = .6 = 60% .2 0 10 6

6. The Most Important Continuous R.V. Consider the function that is defined for every number from (-∞,∞) What is the Area Under This Curve? How many Normal Distributions exist? Infinite Amount 1 = 100% What is the Symmetry/Skewness? For each combination of mu and sigma, there is a unique normal distribution Symmetric

7. The Most Important Continuous R.V. Don’t memorize the complicated formula: If mu and sigma (mean and standard deviation) are given to you then you should know exactly what the distribution looks like: The distribution is CENTERED at MU The distribution is more SPREAD OUT for larger values of SIGMA

8. The Most Important Continuous R.V. Don’t memorize the complicated formula: If mu and sigma (mean and standard deviation) are given to you then you should know exactly what the distribution looks like: The distribution is CENTERED at MU The distribution is more SPREAD OUT for larger values of SIGMA

9. The Most Important Continuous R.V. Don’t memorize the complicated formula: If mu and sigma (mean and standard deviation) are given to you then you should know exactly what the distribution looks like: The distribution is CENTERED at MU The distribution is more SPREAD OUT for larger values of SIGMA

10. The Most Important Continuous R.V. Don’t memorize the complicated formula: If mu and sigma (mean and standard deviation) are given to you then you should know exactly what the distribution looks like: The distribution is CENTERED at MU The distribution is more SPREAD OUT for larger values of SIGMA

11. The Most Important Continuous R.V. Don’t memorize the complicated formula: If mu and sigma (mean and standard deviation) are given to you then you should know exactly what the distribution looks like: The distribution is CENTERED at MU The distribution is more SPREAD OUT for larger values of SIGMA

12. The Most Important Continuous R.V. Don’t memorize the complicated formula: If mu and sigma (mean and standard deviation) are given to you then you should know exactly what the distribution looks like: The distribution is CENTERED at MU The distribution is more SPREAD OUT for larger values of SIGMA

13. The Most Important Continuous R.V. Example: Brown University gives a standardized exam to all students interested in joining the MBA program. The exam is graded in such a way that the scores are normally distributed with a mean of 100 and standard deviation of 10. “Empirical Rule”: For normal distributions, 68% of the values are within 1 standard deviation of the mean, 95% of the values are within 2 standard deviation of the mean, 99.8% of the values are within 3 standard deviation of the mean 90 110

14. The Most Important Continuous R.V. Example: Brown University gives a standardized exam to all students interested in joining the MBA program. The exam is graded in such a way that the scores are normally distributed with a mean of 100 and standard deviation of 10. “Empirical Rule”: For normal distributions, 68% of the values are within 1 standard deviation of the mean, 95% of the values are within 2 standard deviation of the mean, 99.8% of the values are within 3 standard deviation of the mean 80 90 110 120

15. The Most Important Continuous R.V. Example: Brown University gives a standardized exam to all students interested in joining the MBA program. The exam is graded in such a way that the scores are normally distributed with a mean of 100 and standard deviation of 10. “Empirical Rule”: For normal distributions, 68% of the values are within 1 standard deviation of the mean, 95% of the values are within 2 standard deviation of the mean, 99.8% of the values are within 3 standard deviation of the mean 70 80 90 110 120 130

16. Calculations with the Normal Distribution Recall the last example: Average test score (mu) is 100 with 10 as sigma. What’s the probability of scoring over 105? P[X ≥ 105] cannot be found by simple geometry, so we have to do something else – a “Z calculation” – and look up the probability on a table 105

17. Calculations with the Normal Distribution Recall the last example: Average test score (mu) is 100 with 10 as sigma. What’s the probability of scoring over 105? Step 1: Transform the variable X, N(mu, sigma) to Z, N(0,1) .5

18. Calculations with the Normal Distribution Recall the last example: Average test score (mu) is 100 with 10 as sigma. What’s the probability of scoring over 105? Step 1: Transform the variable X, N(mu, sigma) to Z, N(0,1) Step 2: Look up the values in a “z table” (page A58, A59) .5

19. For any value k of Z, you can look up the probabity P[Z≤k] We are interested in the 6th row (0.5) and the first column, since the second decimal point is 0 (0.50) Therefore P[Z ≤ 0.50] = .6915

20. Calculations with the Normal Distribution Recall the last example: Average test score (mu) is 100 with 10 as sigma. What’s the probability of scoring over 105? Step 1: Transform the variable X, N(mu, sigma) to Z, N(0,1) Step 2: Look up the values in the a “z table” (page A58, A59) P[Z ≤ 0.5] = .6915 P[Z > 0.5] = .3085 .5