Chapter4.6-4.7 lecture notes

# Chapter4.6-4.7 lecture notes - Chapter 4.6 and 4.7 4.6 The...

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Chapter 4.6 and 4.7 4.6. The Poisson Approximation to the Binomial Distribution 1. Poisson Distribution f(x)= λ x e - λ /x! for x= 0, 1, 2,…….. = 0 otherwise. Table 2 gives the cumulative distribution of Poisson probabilities (F(x)= = x i 0 λ i e - λ /i! for λ =0.02, 0.04, …0.10,0.15… 1.0, 1.1,1.2,…..6.0, 6.2,6.4….8.0, 8.5,..10.0. 2. Mean of Poisson distribution: μ = λ , i.e. = 0 x x λ x e - λ / x! = λ 3. Variance of Poisson distribution : σ 2 = λ . Standard deviation of a Poisson distribution, σ = √λ i.e. = 0 ( x x - μ ) 2 λ x e - λ / x! = = 0 ( x x - λ ) 2 λ x e - λ / x! = λ Example: The number of weekly breakdowns of a computer is a random variable having a Poisson distribution with λ =0.3. Find the probability that the computer will operate without a breakdown for 1 week? Determine the mean and variance of X=#breakdowns per week. 4. Binomial distribution with large n (n 20) AND small p (p 0.05) approximates Poisson distribution with λ =np. That is, b(x;n,p ) 2245 (np) x e -np /x! for n 20 and p 0.05 EXAMPLE : The chance that a given unit of product is defective is 0.002.
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