MATH 105: Finite Mathematics 9-6: The Normal Distribution Prof. - PowerPoint PPT Presentation

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion MATH 105: Finite Mathematics 9-6: The Normal Distribution Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Outline Bernoulli Probability and the Normal Distribution 1 Properties of the Normal Distribution 2 Examples 3 Conclusion 4

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Outline Bernoulli Probability and the Normal Distribution 1 Properties of the Normal Distribution 2 Examples 3 Conclusion 4

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Graphing Bernoulli Probability Distributions We start this section by examining the “Probability Distribution” we get from a Bernoulli Process. This is called the Binomial Probability Distribution. Example Construct a probability histogram for a Bernoulli process with n = 6 trials where p = 1 2

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Graphing Bernoulli Probability Distributions We start this section by examining the “Probability Distribution” we get from a Bernoulli Process. This is called the Binomial Probability Distribution. Example Construct a probability histogram for a Bernoulli process with n = 6 trials where p = 1 2 r Pr[ r ] ” 0 “ ” 6 ≈ 0 . 01563 “ 1 1 0 C (6 , 0) 2 2 ” 1 “ ” 5 ≈ 0 . 09375 “ 1 1 1 C (6 , 1) 2 2 ” 2 “ ” 4 ≈ 0 . 23438 “ 1 1 2 C (6 , 2) 2 2 ” 3 “ ” 3 ≈ 0 . 31250 “ 1 1 3 C (6 , 3) 2 2 ” 4 “ ” 2 ≈ 0 . 23438 “ 1 1 4 C (6 , 4) 2 2 ” 5 “ ” 1 ≈ 0 . 09375 “ 1 1 5 C (6 , 5) 2 2 ” 6 “ ” 0 ≈ 0 . 01563 “ 1 1 6 C (6 , 6) 2 2

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Graphing Bernoulli Probability Distributions We start this section by examining the “Probability Distribution” we get from a Bernoulli Process. This is called the Binomial Probability Distribution. Example Construct a probability histogram for a Bernoulli process with n = 6 trials where p = 1 2 r Pr[ r ] ” 0 “ ” 6 ≈ 0 . 01563 “ 1 1 0 C (6 , 0) 2 2 ” 1 “ ” 5 ≈ 0 . 09375 “ 1 1 1 C (6 , 1) 2 2 ” 2 “ ” 4 ≈ 0 . 23438 “ 1 1 2 C (6 , 2) 2 2 ” 3 “ ” 3 ≈ 0 . 31250 “ 1 1 3 C (6 , 3) 2 2 ” 4 “ ” 2 ≈ 0 . 23438 “ 1 1 4 C (6 , 4) 2 2 ” 5 “ ” 1 ≈ 0 . 09375 “ 1 1 5 C (6 , 5) 2 2 ” 6 “ ” 0 ≈ 0 . 01563 “ 1 1 6 C (6 , 6) 2 2

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion As n Increases. . . As n increases, one could almost say the number of successes becomes more like a continuous variable. The picture for n = 100 is shown below.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion As n Increases. . . As n increases, one could almost say the number of successes becomes more like a continuous variable. The picture for n = 100 is shown below.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion The Normal Distribution As n continues to increase, the Binomial Distribution approaches the Normal Distribution which is shown below.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion The Normal Distribution As n continues to increase, the Binomial Distribution approaches the Normal Distribution which is shown below.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Outline Bernoulli Probability and the Normal Distribution 1 Properties of the Normal Distribution 2 Examples 3 Conclusion 4

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion General Properties Properties of the Normal Distribution The following are properties of the Normal Distribution. Bell shaped Symmetric about µ Probability = Area Area Under Curve = 1 Probability a value lies between a and b is area under curve between a and b . Standard normal distribution has µ = 0 and σ = 1.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion General Properties Properties of the Normal Distribution The following are properties of the Normal Distribution. Bell shaped Symmetric about µ Probability = Area Area Under Curve = 1 Probability a value lies between a and b is area under curve between a and b . Standard normal distribution has µ = 0 and σ = 1.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion General Properties Properties of the Normal Distribution The following are properties of the Normal Distribution. Bell shaped Symmetric about µ Probability = Area Area Under Curve = 1 Probability a value lies between a and b is area under curve between a and b . Standard normal distribution has µ = 0 and σ = 1.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion General Properties Properties of the Normal Distribution The following are properties of the Normal Distribution. Bell shaped Symmetric about µ Probability = Area Area Under Curve = 1 Probability a value lies between a and b is area under curve between a and b . Standard normal distribution has µ = 0 and σ = 1.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion General Properties Properties of the Normal Distribution The following are properties of the Normal Distribution. Bell shaped Symmetric about µ Probability = Area Area Under Curve = 1 Probability a value lies between a and b is area under curve between a and b . Standard normal distribution has µ = 0 and σ = 1.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion General Properties Properties of the Normal Distribution The following are properties of the Normal Distribution. Bell shaped Symmetric about µ Probability = Area Area Under Curve = 1 Probability a value lies between a and b is area under curve between a and b . Standard normal distribution has µ = 0 and σ = 1.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion General Properties Properties of the Normal Distribution The following are properties of the Normal Distribution. Bell shaped Symmetric about µ Probability = Area Area Under Curve = 1 Probability a value lies between a and b is area under curve between a and b . Standard normal distribution has µ = 0 and σ = 1.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Empirical Rule Empirical Rule In a normal distribution, approximately 1 68% of outcomes within 1 standard deviation of the mean. 2 95% of outcomes within 2 standard deviations of the mean. 3 99.7% of outcomes within 3 standard deviations of the mean.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Empirical Rule Empirical Rule In a normal distribution, approximately 1 68% of outcomes within 1 standard deviation of the mean. 2 95% of outcomes within 2 standard deviations of the mean. 3 99.7% of outcomes within 3 standard deviations of the mean.

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Applying the Empirical Rule Example A standardized test has a mean score of µ = 125 and a standard deviation of σ = 12. 1200 students take the test. 1 How many scored between 113 and 137? 2 How many scored above 125? 3 How many students scored less than 89?

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Applying the Empirical Rule Example A standardized test has a mean score of µ = 125 and a standard deviation of σ = 12. 1200 students take the test. 1 How many scored between 113 and 137? 2 How many scored above 125? 3 How many students scored less than 89?

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Applying the Empirical Rule Example A standardized test has a mean score of µ = 125 and a standard deviation of σ = 12. 1200 students take the test. 1 How many scored between 113 and 137? 0 . 68 × 1200 = 816 2 How many scored above 125? 3 How many students scored less than 89?

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion Applying the Empirical Rule Example A standardized test has a mean score of µ = 125 and a standard deviation of σ = 12. 1200 students take the test. 1 How many scored between 113 and 137? (816) 2 How many scored above 125? 3 How many students scored less than 89?