QBM117 Business Statistics

1. QBM117Business Statistics Probability and Probability Distributions Continuous Probability Distributions 1

2. Objectives • To differentiate between a discrete probability distribution and a continuous probability distribution • To introduce the probability density function and its relationship to a continuous random variable • To introduce the uniform distribution • To introduce the normal distribution • To introduce the standard normal random variable • Use the standard normal tables to find probabilities 2

3. Continuous Random Variables • A continuous random variable has an infinite number of possible values. • It can assume any value in an interval. • We cannot list all the possible values of a continuous random variable. 3

4. Continuous Probability Distribution • A continuous random variable has an infinite number of possible values. • It is not possible to calculate the probability that a continuous random variable will take on a specific value. • Instead we calculate the probability that a continuous random variable will lie in a specific interval. 4

5. Probability Density Function • A smooth curve is used to represent the probability distribution of a continuous random variable. • The curve is called a probability density function and is denoted by . 5

6. A probability density function must satisfy two conditions: • is non-negative, • The total area under the curve is equal to 1 6

7. If X is a continuous random variable with probability density function , the probability that X will take a value between a and b, , is given by the area under the curve between a and b. 7

8. The Uniform Distribution • A continuous random variable X, defined over an interval , is uniformly distributed if its probability density function is given by 8

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10. To calculate the probability that X falls between and we need to find the area of the rectangle whose base is and whose height is 10

11. Expected Value and Variance of a Uniform Random Variable • The expected value and variance of a uniform random variable are given by 11

12. Example 1 The total time to process a loan application is uniformly distributed between 3 and 7 days. • Define the probability density function for loan processing time. • What is the probability that a loan will be processed in less than 3 days? • What is the probability that a loan will be processed in 5 days or less? • Find the expected processing time and its standard deviation. 12

13. The probability density function for loan processing time is given by 13

14. The probability that a loan will be processed in less than 3 days: • The probability that a loan will be processed in 5 days or less: 14

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16. The expected processing time: Variance: Standard deviation: 16

17. The Normal Distribution • A continuous random variable X with mean and standard deviation is normally distributed if its probability density function is given by where = 2.71828… and = 3.14159… 17

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19. A normal random variable is normally distributed with mean and standard deviation • The normal distribution is described by two parameters, and . • is the mean and determines the location of the curve. • is the standard deviation and determines the spread about the mean, that is the width of the curve. • There is a different normal distribution for each combination of a mean and a standard deviation. 19

20. m = 10 m = 11 m = 12 s= 2 s =3 s =4 20

21. For the area under the normal curve • 68% lies between and • 95% lies between and • 99.7% lies between and 21

22. Calculating Normal Probabilities • If X is a normal random variable with probability density function , the probability that X will take a value between a and b, , is given by the area under the normal curve between a and b. 22

23. Standard Normal Distribution • There are an infinite number of normal distributions because and can take an infinite number of possible values. • All normal distributions are related to the standard normal distribution. • A standard normal random variable Z is normally distributed with a mean and a standard deviation . • We can convert any normal random variable X to a standard normal random variable Z. 23

24. Standard Normal Probability Table • Areas under the standard normal probability density function have been calculated and are available in Table 3 of Appendix 3 of the text. • This table gives the probability that the standard normal random variable Z lies between 0 and z. 24

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26. Using Table 3 • Always draw a diagram. • Shade in the area representing the probability you are trying to find. 26

27. Example 2 What is the probability of finding a z value • between 0 and 1.43 • between –0.87 and 0 • greater than 2.35 • less than –1.98 • less than 1.06 • greater than –1.30 • between –1.83 and 2.01 • between 0.89 and 2.12 • between –2.15 and –1.68 • less than –1.48 or greater than 1.13 27

28. The probability of finding a z value between 0 and 1.43: 28

29. The probability of finding a z value between –0.87 and 0: 29

30. The normal distribution is symmetric, therefore 30

31. The probability of finding a z value greater than 2.35: 31

32. Since the total area under the normal curve equals one, and since the curve is symmetric about 0, the area to the right of 0 is 0.5. Therefore 32

33. The probability of finding a z value less than –1.98: 33

34. The probability of finding a z value less than 1.06: 34

35. The probability of finding a z value greater than –1.30: 35

36. The probability of finding a z value between –1.83 and 2.01: 36

37. The probability of finding a z value between 0.89 and 2.12: 37

38. The probability of finding a z value between –2.15 and –1.68: 38

39. The probability of finding a z value less than –1.48 or greater than 1.13: 39

40. Reading for next lecture • Chapter 5, reread section 5.7 Exercises • 5.50 • 5.51 • 5.52 40