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Chapter 4b

# Chapter 4b - 4.6 Normal Approximation to the Binomial For a...

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4.6 Normal Approximation to the Binomial For a very large n, we can use the normal distribution to approximate binomial probabilities. What is large? Approaching infinity In reality, if np > 5 and n(1-p) > 5, the binomial approximation can be used. If X be a binomial variable with parameters n and p such that np > 5 and n(1-p) > 5, then X is approximately normally distributed with mean μ =np and variance σ 2 =np(1-p). We can then use the standard normal table to get probabilities. Note: We are going to be modeling a discrete variable using a continuous one, so we need to use a continuity correction factor of .5 when finding our answers. Binomial Situation Normal Approximation P(X > 12) P(X > 11.5) P(X >12) P(X > 12.5) P(X < 12) P(X < 12.5) P(X < 12) P(X < 11.5) P(X = 12) P(11.5 < X < 12.5)

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Example: A particularly long traffic light on your morning commute is green 40% of the time as you approach it. Assume that each morning represents an independent trial. Let X represent the number of mornings the light is green over a 30 day period. Is it appropriate to use the normal approximation? What is the probability that X is less than 20? What is the probability that X is greater than 22? What is the probability that X equals 15?

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Other Continuous Distributions Some random variables are always non-negative and for various reasons yield distributions that are skewed to the right. Such variables include the lengths of time between malfunctions for aircraft engines and the length of time between arrivals at a supermarket checkout line. The Gamma, Exponential, Chi-squared and Weibull are related distributions that can model such situations. 4.3 Gamma Distribution The Gamma distribution is partially based on the gamma function. - - = Γ 0 1 ) ( dz e z z α α where α >0 Properties of the gamma function: (1) 1 ) 1 ( = Γ (2) For a > 1, )! 1 ( ) 1 ( ) 1 ( ) ( - = - Γ - = Γ α α α α Gamma Probability Function β α α β α / 1 ) ( 1 ) ( x e x x f - - Γ = where x ≥ 0, α > 0, β > 0
The shape of the distribution changes based on α and β. We can break the shape down based on the value of α , the shape parameter. Alpha > 1 : This is the most frequently encountered situation, particularly, of course, when α is an integer. The Gamma density function is then an asymetric bell-shaped curve. Alpha = 1: The Gamma distribution is then just an exponential distribution . Alpha < 1: The vertical axis is now an asymptote and the Gamma distribution is monotonously decreasing. These websites visualize how the shape of the distribution changes as the parameters change: http://personal.kenyon.edu/hartlaub/MellonProject/Gamma2.html http://www.aiaccess.net/English/Gloss aries/GlosMod/e_gm_gamm a_distri.htm Observe that α has a larger influence than β on the shape of the curve: β merely stretches the curve horizontally and compresses it vertically. Hence, β is referred to as the scale parameter.

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