1. PROBABILITY DISTRIBUTION

2. Probability Distribution of a Continuous Variable

4. The Normal Distribution “Gaussian Distribution ” Is a theoretical model that has been found to fit many naturally occurring phenomena. Is a theoretical model that has been found to fit many naturally occurring phenomena. It is the most important distribution in statistics It is the most important distribution in statistics It is used for continuous variables It is used for continuous variables

5. The Normal Distribution “Gaussian Distribution ” The parameters in this distribution are the: The parameters in this distribution are the: Population mean ( µ ) as a measure of central tendency Population mean ( µ ) as a measure of central tendency Population standard deviation (σ) as a measure of dispersion Population standard deviation (σ) as a measure of dispersion

6. The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . Normal Probability Distribution Characteristics Standard Deviation  Mean  x

7. The Normal Distribution “Gaussian Distribution” The curve is symmetric around the mean The curve is symmetric around the mean The total area under the curve equal one The total area under the curve equal one

8. The distribution is symmetric, and is bell-shaped. The distribution is symmetric, and is bell-shaped. Normal Probability Distribution Characteristics x

9. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. Normal Probability Distribution Characteristics x

10. The Normal Distribution “Gaussian Distribution” The mean, median, and the mode are equal The mean, median, and the mode are equal Mean=Median=Mode Total P=1

11. Normal Probability Distribution Characteristics -10020 The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. x

12. The Normal Distribution “Gaussian Distribution” 50% of the area under the curve is on the right side of the curve and the other 50% is on its left 50% of the area under the curve is on the right side of the curve and the other 50% is on its left

13. Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Normal Probability Distribution Characteristics.5.5 x

14. The Normal Distribution “Gaussian Distribution” With fixed (σ) different values of µ will shift the graph of the distribution along the X axis With fixed (σ) different values of µ will shift the graph of the distribution along the X axis The shape of the curve will not changed, but it will be shifted to: The shape of the curve will not changed, but it will be shifted to: the right ( when µ is increased) the right ( when µ is increased) or to the left (when µ is decreased) or to the left (when µ is decreased)

15. 8.15 Normal Distribution…

16. Normal Probability Distribution Characteristics  = 15  = 25 The standard deviation determines the width of the curve: larger values result in wider, flatter curves. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x

17. The Normal Distribution “Gaussian Distribution” Different values of (σ) determine the degree of flatness or peakedness of the graph of the distribution Different values of (σ) determine the degree of flatness or peakedness of the graph of the distribution When (σ) is increased the curve will be more flat When (σ) is increased the curve will be more flat When (σ) is decreased the curve will be more peaked When (σ) is decreased the curve will be more peaked

18. 8.18 Normal Distribution…

20. The Normal Distribution “Gaussian Distribution” µ ± 1 σ 68% of the area µ ± 1 σ 68% of the area µ ± 2 σ 95% of the area µ ± 2 σ 95% of the area µ ± 3 σ 99.7% of the area µ ± 3 σ 99.7% of the area

21. The Normal Distribution “Gaussian Distribution” µ ± 1 σ 68% of the area µ ± 1 σ 68% of the area

22. The Normal Distribution “Gaussian Distribution” µ ± 2 σ 95% of the area µ ± 2 σ 95% of the area

23. The Normal Distribution “Gaussian Distribution” µ ± 3 σ 99.7% of the area µ ± 3 σ 99.7% of the area

24. 68-95-99.7 Rule 68% of the data 95% of the data 99.7% of the data

25. The unit normal, or the Standard normal distribution X- µ X- µ Z= --------- Z= --------- σ

26.  0 z The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. Standard Normal Probability Distribution

28. Exercise Find for a standard normal distribution a) P(0< Z <1.2) b) P(Z >1.2) c) P(-1.2< Z <1.2) d) P(Z 1.2) e) P(Z <1.2) f) P(1.5 < Z <2.0)

30. Exercise If µ of DBP of a population = 80 mmHg, and σ 2 =100(mmHg) 2.What is the probability of selecting a man with DBP of: 1) P(75< X < 85) 2) P(60< X <100) 3) P(65< X <95) 4) P(X <60) 5) P(X >100) 6) P(90< X <100)

31. 80 mmHg 10 mmHg X 90100110 506070

33. Exercise If the weight of 6-years old boys is normally distributed with µ =25 Kg, and σ = 2 kg. Find: 1. P(20< X <25) 2. P(X >28) 3. P(X >22) 4. P(X <22) 5. P(X <28) 6. P(26< X <29)

34. 25Kg 2 Kg X 272931 192123