1. 4. Random Variables A random variable is a way of recording a quantitative variable of a random experiment.

2. 4. Random Variables A random variable is a way of recording a quantitative variable of a random experiment. In particular Chapter 4 talks about discrete random variables.

3. 4. Random Variables A random variable is a way of recording a quantitative variable of a random experiment. In particular Chapter 4 talks about discrete random variables. If a random variable has a particular distribution (such as a binomial distribution) then our work becomes easier. We use formulas and tables.

4. 5. Continuous Random Variables A random variable is a way of recording a quantitative variable of a random experiment. In particular Chapter 4 talks about discrete random variables. If a random variable has a particular distribution (such as a binomial distribution) then our work becomes easier. We use formulas and tables.

5. 5. Continuous Random Variables A random variable where X can take on a range of values, not just particular ones. Examples: Heights Distance a golfer hits the ball with their driver Time to run 100 meters Electricity usage of a home.

6. Continuous probability distribution functions For a discrete random variable, probabilities are given as a table of values, and the distribution can be graphed as a bar graph. For a continuous random variable, probabilities are specified by a continuous function. The graph of the probability distribution function is a curve.

7. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 5.1 A probability f(x) for a continuous random variable x

8. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Definition

9. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 5.2 Density Function for Friction Coefficient, Example 5.1

10. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Find probability friction is less than 10.

11. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Find probability friction is less than 10. Solution:Probabiity = area of shaded triangle = (1/2)(5)(0.2)=0.5

12. Uniform Distribution A Uniform Distribution has equally likely values over the range of possible outcomes.

13. Uniform Distribution A Uniform Distribution has equally likely values over the range of possible outcomes. A graph of the uniform probability distribution is a rectangle with area equal to 1.

14. Example The figure below depicts the probability distribution for temperatures in a manufacturing process. The temperatures are controlled so that they range between 0 and 5 degrees Celsius, and every possible temperature is equally likely. x P(x) Temperature (degrees Celsius) 0 1 2 3 4 5 0.2 0

15. Example Note that the total area under the “curve” is 1. x P(x) Temperature (degrees Celsius) 0 1 2 3 4 5 0.2 0

16. Example What is the Probability that the temperature is exactly 4 degrees? x P(x) Temperature (degrees Celsius) 0 1 2 3 4 5 0.2 0

17. Example What is the Probability that the temperature is exactly 4 degrees? Answer: 0 x P(x) Temperature (degrees Celsius) 0 1 2 3 4 5 0.2 0

18. Since we have a continuous random variable there are an infinite number of possible outcomes between 0 and 5, the probability of one number out of an infinite set of numbers is 0. Explanation

19. Example What is the probability the temperature is between 1 0 C and 4 0 C? x P(x) Temperature (degrees Celsius) 0 1 2 3 4 5 0.2 0

20. Example What is the probability the temperature is between 1 0 C and 4 0 C? x P(x) Temperature (degrees Celsius) 0 1 2 3 4 5 0.2 0

21. What is the probability the temperature is between 1 0 C and 4 0 C? We know that the total area of the rectangle is 1, and we can see that the part of the rectangle between 1 and 4 is 3/5 of the total, so P(1  x  4) = 3/5*(1) = 0.6. x P(x) Temperature (degrees Celsius) 0 1 2 3 4 5 0.2 0

22. Review: Probabilities and Area For a density curve depicting the probability distribution of a continuous random variable, –the total area under the curve is 1, –there is a direct correspondence between area and probability. –Only the probability of an event occurring in some interval can be evaluated. –The probability that a continuous random variable takes on any particular value is zero.

23. General Uniform Distribution A Uniform Distribution has equally likely values over the range of possible outcomes, say c to d.

24. Normal Distributions This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bell- shaped curves.

25. Normal Distributions This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bell- shaped curves. Reminder: Mu is the mean, sigma is the standard deviation.

26. Examples The following are examples of normally distributed everyday data. –Grades on a test. –How many chips are in a small bag of potatoe chips. –The measurements of distance between two points. –The heights of students in this class.

27. Normal Distributions

28. Shape of this curve is determined by µ and σ –µ it’s centered, σ is how far it’s spread out.

29. Standard Normal Distribution The Standard Normal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.  In this way the formula giving the heights of the normal curve is simplified greatly.

30. Z-score

31. Standard Normal Probabilities P(0  z  1) represents the probability that z takes on values between 0 and 1, which is represented by the area under the curve between 0 and 1. P(0  z  1) = 0.3413

32. P(0  z  1) = 0.341 Revelation! Since the mean is 0 and the standard deviation is 1, this tells us that the probability that z is within one standard deviation of the mean (either below or above) is (2)(0.341)= 0.682.

33. P(0  z  1) = 0.341 Revelation! Since the mean is 0 and the standard deviation is 1, this tells us that the probabiity that z is within one standard deviation of the mean (either below or above) is (2)(0.341)= 0.682. Agrees with Empirical Rule: 68% of data lies within one standard deviation of the mean

34. Finding Probabilities when given z-scores. For a given z-score, the probability can be found in a table in the back of the text (Table IV), also see inside front cover. Note: The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks.

35. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Table 5.1

36. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Find probability z is between - 1.33 and +1.33.

37. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Want probability z is between -1.33 and +1.33. Solution: Locate 1.33 in the row labeled 1.3 and the column labeled.03. By symmetry, ans = 2(0.4082) =.8164

38. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Find probability z exceeds 1.96 in absolute value.

39. Copyright © 2013 Pearson Education, Inc.. All rights reserved. Areas under the standard normal curve for z exceeding 1.96 in absolute value

41. Revelation! It follows that the area of the un- shaded region is 0.95. Agrees with Empirical Rule which states that, for data sets having a mound shaped distribution, 95% of the values lie within approximately 2 standard deviations of the mean

42. Keys to success Learn the standard normal table and how to use it. We will be using these tables through out the course.