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Facts about Binomial & Poisson distribution Fact 1 : If X ~ Bin (n, p), then E ( X ) np var( X ) np(1 p) Fact 2 : If X 1 ~ Bin (n1 , p), X 2 ~ Bin (n2 , p) independen tly, then X 1 X 2 ~ Bin (n1 n2 , p) Fact 3 : If X ~ Poisson ( ), then E ( X ) var( X ) Normal Approximation to Binomial Distribution p= 0.4, n=5 0.3 p= 0.4, n=10 p= 0.4, n=25 0.2 0.1 0.2 0.1 0.1 0 1 2 3 4 5 Bin (n, p) N np, np(1 p) if n is large. Demo: http://www.ruf.rice.edu/~lane/stat_sim/index.html Normal Approximation to Poisson Distribution Poisson ( ) N , if n is large. Topic 8: Normal Distribution A continuous random variable is one that can take on any value within an interval. The distribution of a continuous variable X is given by a probability density function f (x) satisfying (i) f ( x) 0 for all x (,) b (ii) P(a X b) f ( x)dx a (iii) f ( x)dx 1. Note that it is the integral of f ( x), i.e., the area under the density curve, and not f ( x) itself, that gives us the probabilit y In particular , P( X x) 0 f ( x) for all x Can think of the probability density f(x) as the relative frequency histogram of a very large sample Expectation defined similarly as in discrete but with integral instead of summation: E( X ) x f ( x)dx Can again interpret expected value E(X) as the long-run average of X under repeated sampling The most well known continuous distribution is the normal distribution. Standard normal density 1 z2 2 1 z ( z) e , 2 Bell-shaped Symmetric about 0 Mean = 0, variance = SD = 1 Well tabulated From N (0,1) to N ( , 2 ) and vice versa If Z ~ N (0,1) , then X Z ~ N ( , 2 ) 2 Conversely, if X ~ N ( , ) , then X Z ~ N (0,1) (Standardization) Normally distributed random variables should be standardized before looking up table X Blood Pressure ~ N ( 129 mm Hg , 19.8 ) 2 2 X 129 150 129 P( X 150) P P( Z 1.06) 0.145 19.8 19.8 Demo: http://www.isds.duke.edu/sites/java.html The standard normal density curve If X ~ N ( , ) 2 X P( | X | 2 ) P Z 2 within 2 SD from the mean 0.954 P( | X | 3 ) 0.997 within 3 SD from the mean This is the empirical rule Normal distribution is often used to model continuous measurement data such as weight, height, blood pressure, etc. The use of normal distribution is often justified by the Central Limit Theorem which says that the sum/average of a large number of independent and identically distributed variables is approximately normally distributed. Measurement error = sum of many indep smaller errors IQ : determined by many genetic & environmental factors Demo: http://www.ruf.rice.edu/~lane/stat_sim/index.html