Chapter 3 Continuous Random Variables

1. Chapter 3 Continuous Random Variables

2. 3.2 – Definitions Definition 3.2.1 A random variable X is said to be continuous if there exists a function f called the probability density function (p.d.f ) which is continuous at all but a finite number of points and satisfies the following three properties:

3. The c.d.f. • The cumulative distribution function (c.d.f.) of a continuous random variable X is • An important relationship:

4. Example 3.2.1 Suppose that X is a continuous random variable with range [1, 2] and p.d.f

5. Example 3.2.1 • Calculate : • Find the c.d.f.:

6. Definition 3.2.2 Let X be a continuous random variable with p.d.f. f (x)

7. Example 3.2.2

8. Percentiles Definition 3.2.3 Let X be continuous random variable with p.d.f. f and c.d.f. F, and let p be a number between 0 and 1. The (100p)th percentile is a value of X, denoted πp, such that • The 50thpercentile is called the median and is denoted m. • The mode of X is the value x for which f is maximum.

9. Special Percentiles • 25th – First Quartile – p1 =π0.25 • 50th– Second Quartile – p2=π0.50 = m • 75th– Third Quartile – p3=π0.75

10. Example 3.2.4

11. 3.3 – Uniform and Exponential Definition 3.3.1 A continuous random variable X has a uniform distribution if its p.d.f. is where a < b are any real numbers. The phrases “X is U(a, b)” and “X is uniformly distributed over [a, b]” mean that X has this type of distribution.

12. Uniform Distribution

13. Example 3.3.2 The owner of a local gas station observes that customers typically purchase between five and 20 gallons of gasoline at each fill-up and that no amount is more frequent than any other. If gas costs $2.50 per gallon, find the probability that a randomly selected customer spends more than $30 on gas.

14. Example 3.3.2 Let X = gallons of gas purchased by a customer • Assume X is U(5, 20) • for Let Y = amount of money spent by a customer • Y = 2.5X

15. Exponential Distribution Definition 3.3.2 A random variable X has an exponential distribution if its p.d.f. is where • Describes waiting time between events that are described by a Poisson distribution

16. Exponential Distribution

17. Example 3.3.6 Suppose a manufacturer of electrical wire observes a mean of 0.5 defects per 100 ft of wire. Find the mean distance between defects (in ft) and find the probability that the distance between one defect and the next is less than 125 ft. • Assume the number of defects per ft has a Poisson distribution

18. Example 3.3.6 Let X = the distance in ft between defects • “Wait time” between defects • X is exponential with

19. 3.4 – The Normal Distribution Definition 3.4.1 A continuous random variable Z has a standard normal distribution if its p.d.f. is

20. The c.d.f. See Table C.1 – Two important properties Example 3.4.1

21. The Normal Distribution Definition 3.4.2 A continuous random variable X has a normal distribution if its p.d.f. is where μ is any real number and σ > 0 • The phrase “X is ” means that X has a normal distribution with parameters μ and σ

22. The Normal Distribution

23. Empirical Rule

24. Relationship Theorem 3.4.1 If X isthen the random variable is Proof: We find the c.d.f. of Z:

25. Let so that , Thus so Z has the same c.d.f. as that of a standard normal random variable. Therefore, it must have the same p.d.f., and we conclude that Z is N (0, 1).

26. Application The “z-score” of x is

27. Example 3.4.2 Consider the random experiment of choosing a woman at random and measuring her height. Let the random variable X be the height in inches. Assuming that X is , find the probability of selecting a woman who is between 60 and 62 inches tall.

28. Example 3.4.2 We want • Calculate z-scores:

29. 3.5 – Functions of Continuous Random Variables Goal: To find the p.d.f. of a variable defined in terms of another variable • An important relationship • Second Fundamental Theorem of Calculus

30. Example 3.5.2 Consider the random variable X with range and p.d.f. . Define the variable . Find the p.d.f. of Y. • Find the c.d.f. • Find the p.d.f.

31. Simulations Theorem 3.5.1 Let Y be a random variable that is and let satisfy the requirements of a c.d.f. of a continuous random variable with certain properties. • Letdenote the inverse of the function • Define the random variable • Then has the c.d.f.

32. Example 3.5.4 Generate values of a random variable X with an exponential distribution • Find the c.d.f.: • Find

33. Example 3.5.4 • Use software to generate values of a random variable Y that is • Suppose • - Values of Y • - Values of X

34. 3.6 – Joint Distributions Definition 3.6.2 Let X and Y be two continuous random variables. The joint probability density function, or joint p.d.f., is an integrable function f(x, y) that satisfies the following three properties:

35. Marginal Distributions The marginal probability density functions, or marginal p.d.f.’s, of X and Y are Definition 3.6.3 Two discrete or continuous random variables X and Y are said to be independent if

36. Example 3.6.3 Suppose the joint p.d.f. of X and Y is Note Marginal p.d.f.’s:

37. Example 3.6.4 Suppose a man and a woman agree to meet at a restaurant sometime between 6:00 and 6:15 PM. Find the probability that the man has to wait longer than five minutes for the woman to arrive. • Let X and Y denote the number of minutes past 6:00 PM that the man and the woman arrive, respectively.

38. Example 3.6.4 Assume Then

39. 3.7 – Functions of Independent Random Variables Definition 3.7.1 The mean or expected value of a function of two random variables and , is where is the joint p.d.f. of and

40. Example 3.7.1 Suppose the joint p.d.f. of X and Yis

41. Properties Theorem 3.7.1 For any discrete or continuous random variables, If X and Y are independent, then Theorem 3.7.2 If X and Yare independent, then

42. Properties Theorem 3.7.3 Ifandare independent random variables with respective m.g.f.’s and , and a and b are constants, then the m.g.f. of is

43. Properties

44. Example 3.7.5 Let X and Y be independent and both Define . Then is So that

45. 3.8 – The Central Limit Theorem Definition 3.8.1 When a set of objects is selected from a larger set of objects, the larger set is called the population and the smaller set is called the sample. The number of objects in the sample is called the sample size.

46. The Central Limit Theorem Theorem 3.8.2 If X1, X2,…, Xnare mutually independent random variables with a common distribution, mean μ, and variance σ2, then as the distribution of the sample mean

47. Example 3.8.5 Suppose people at a certain movie theater wait a mean of 30 sec with a variance of 52 sec2 to buy tickets. Find the probability that the mean wait time of a group of 50 movie-goers is less than 28 sec.

48. Example 3.8.5 By CLT: Then

49. 3.9 – The Gamma and Related Distributions Definition 3.9.2 A random variable X has a gamma distribution with parameters r and λ where and if its p.d.f. is

50. Chi-Square Distribution Theorem 3.9.1 Let be independent random variables each with a standard normal distribution. Define the random variable Definition 3.9.3 A random variable with this p.d.f. is said to have a chi-square distribution with n degrees of freedom.