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Normal distribution and intro to continuous probability density functions... Discrete Distribution Symmetrical Binomial Distribution B(10, 0.5) 0.3 P(X=r) 0.25 Prob 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 r Mean = np = 10 x 0.5 = 5 8 9 10 As a Histogram (Area of rectangle = probability) Symmetrical Binomial Distribution B(10, 0.5) 0.3 P(X=r) 0.25 Prob 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 r 6 7 8 9 10 Decrease interval size... Symmetrical Binomial Distribution B(30, 0.5) 0.16 P(X=r) 0.14 0.12 Prob 0.1 0.08 0.06 0.04 0.02 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 r Decrease interval size more…. 0.09 Binomial Distribution : B(100,0.5) P(X=r) 0.08 0.07 0.05 Almost a nice continuous curve 0.04 0.03 0.02 0.01 r 95 10 0 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 0 Prob 0.06 Continuous probability density functions • The curve describes probability of getting any range of values, say P(X > 60), P(X<30), P(20 < X < 50) • Area under the curve = probability • Area under whole curve = 1 • Probability of getting specific number is 0, e.g. P(X=60) = 0 Characteristics of normal distribution • Symmetric, bell-shaped curve. • Shape of curve depends on population mean and variance 2. • Center of distribution is . • Spread is determined by . • Most values fall around the mean, but some values are smaller and some are larger. • Probabilities are from area under the curve The Normal Distribution WRITTEN : X ~ N ( , ) 2 … which means the continuous random variable X is normally distributed with mean and variance 2 Examples of normal random variables Probability student scores higher than 75? 0.08 0.07 Density 0.06 0.05 P(X > 75) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 Grades 75 80 85 Properties of Normal Distribution