4-2 Probability Distributions and Probability Density Functions. Figure 4-2 Probability determined from the area under f(x).

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3 4-2 Probability Distributions and Probability Density Functions Figure 4-2 Probability determined from the area under f(x).

4 4-2 Probability Distributions and Probability Density Functions Definition

5 4-2 Probability Distributions and Probability Density Functions Figure 4-3 Histogram approximates a probability density function.

6 4-2 Probability Distributions and Probability Density Functions

7 4-2 Probability Distributions and Probability Density Functions Example

8 b) What is the probability that a metal cylinder has a diameter between 49.8 mm and 50.1 mm?

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11 4-2 Probability Distributions and Probability Density Functions Example 4-2

12 4-2 Probability Distributions and Probability Density Functions Figure 4-5 Probability density function for Example 4-2.

13 4-2 Probability Distributions and Probability Density Functions Example 4-2 (continued)

14 4-3 Cumulative Distribution Functions Definition

15 4-3 Cumulative Distribution Functions Example

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17 4-3 Cumulative Distribution Functions Example 4-4

18 4-3 Cumulative Distribution Functions Figure 4-7 Cumulative distribution function for Example 4-4.

19 4-4 Mean and Variance of a Continuous Random Variable Definition

20 4-4 Mean and Variance of a Continuous Random Variable Example 4-6

21 4-4 Mean and Variance of a Continuous Random Variable Example 4-8

22 Example Suppose that the diameter of a metal cylinder has a pdf 2 of: f f x x 50 x 0 elsewhere What is the expected value of the cylinder diameter? for 49.5 x 50.5

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24 4-5 Continuous Uniform Random Variable Definition

25 4-5 Continuous Uniform Random Variable Figure 4-8 Continuous uniform probability density function.

26 4-5 Continuous Uniform Random Variable Mean and Variance

27 4-5 Continuous Uniform Random Variable Example 4-9

28 4-5 Continuous Uniform Random Variable Figure 4-9 Probability for Example 4-9.

29 4-5 Continuous Uniform Random Variable

30 4-6 Normal Distribution Definition

31 4-6 Normal Distribution Figure 4-10 Normal probability density functions for selected values of the parameters and 2.

32 4-6 Normal Distribution Some useful results concerning the normal distribution

33 4-6 Normal Distribution Definition : Standard Normal

34 4-6 Normal Distribution Example 4-11 Figure 4-13 Standard normal probability density function.

35 4-6 Normal Distribution Standardizing

36 4-6 Normal Distribution Example 4-13

37 4-6 Normal Distribution Figure 4-15 Standardizing a normal random variable.

38 4-6 Normal Distribution To Calculate Probability

39 4-6 Normal Distribution Example 4-14

40 4-6 Normal Distribution Example 4-14 (continued)

41 4-6 Normal Distribution Example 4-14 (continued) Figure 4-16 Determining the value of x to meet a specified probability.

42 4-7 Normal Approximation to the Binomial and Poisson Distributions Under certain conditions, the normal distribution can be used to approximate the binomial distribution and the Poisson distribution.

43 4-7 Normal Approximation to the Binomial and Poisson Distributions Figure 4-19 Normal approximation to the binomial.

44 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-17

45 4-7 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Binomial Distribution

46 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-18

47 4-7 Normal Approximation to the Binomial and Poisson Distributions Figure 4-21 Conditions for approximating hypergeometric and binomial probabilities.

48 4-7 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Poisson Distribution

49 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-20

50 4-8 Exponential Distribution Definition

51 4-8 Exponential Distribution More Explanation on the exponential distribution

52 4-8 Exponential Distribution Proof for the pdf of the exponential distribution Note: The derivation of the distribution of X depends only on the assumption that the flaws in the wire follow a Poisson Distribution.

53 4-8 Exponential Distribution Mean and Variance

54 4-8 Exponential Distribution Example 4-21

55 4-8 Exponential Distribution Figure 4-23 Probability for the exponential distribution in Example 4-21.

56 4-8 Exponential Distribution Example 4-21 (continued)

57 4-8 Exponential Distribution Example 4-21 (continued)

58 4-8 Exponential Distribution Example 4-21 (continued)

59 4-11 Lognormal Distribution

60 4-11 Lognormal Distribution Figure 4-27 Lognormal probability density functions with = 0 for selected values of 2.

61 4-11 Lognormal Distribution Example 4-26

62 4-11 Lognormal Distribution Example 4-26 (continued)

63 4-11 Lognormal Distribution Example 4-26 (continued)

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