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ENGR 224/STAT 224 Probability and Statistics Lecture 11

ENGR 224/STAT 224 Probability and Statistics Lecture 11. Homework Problems. Problem 36 Problem 38. Normal Approximation to Binomial Distribution. The arithmetic involved in calculating binomial distributions can become immense,

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ENGR 224/STAT 224 Probability and Statistics Lecture 11

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  1. ENGR 224/STAT 224 Probability and StatisticsLecture 11

  2. Homework Problems • Problem 36 • Problem 38

  3. Normal Approximation to Binomial Distribution • The arithmetic involved in calculating binomial distributions can become immense, • fortunately, we can approximate the binomial probability with a normal distribution when the binomial distribution is not too skewed, • Usually used for n greater than 15.

  4. Example: b(7,15;0.4) • Let n=15, p=0.4 Find P(X=7). • Using the table b(7,15;0.4)=.177 • Using the binomial formula

  5. Example: b(7,15;0.4) • Using a normal approximation. We need m=npand s=npq. • Continuity Correction Factor (± 0.5) • m = 0.4 * 15 = 6 • s = 0.4*0.6*15 = 1.897 • Error = .0056

  6. Normal Approximation to Binomial Probabilities • In general if n is large and p is not too close to one or zero, (i.e., if np 10 and nq 10) then

  7. Example Among Canadian households, 24% have telephone answering machines (based on a recent survey). If a telemarketing campaign involves 2500 households, find the probability that at least 650 have answering machines. Solution

  8. Definition: Percentiles • Let p be a number between 0 and 1, then the (100p)th percentile of the probability distribution f(x) is the number h(p) such that the area under the curve from negative infinity to h(p) is p. The median is the 50th percentile

  9. Example • Let X~N(100,25). Determine the 10th percentile for this random variable.

  10. Definition: Exponential Distribution • Exponential PMF • E(X) = 1/l • V(X)=1/l2 • Exponential CDF

  11. Example • Calculate the median of an exponential Distribution

  12. Definition: Gamma Function • Properties: • For a>1, G(a)=(a-1)G(a-1) • For any positive integer, n, G(n)=(n-1)! • G(1/2)=√p

  13. Definition: Gamma Distribution • Gamma PMF • E(X) = ab • V(X)=ab2 • The standard Gamma distribution has b =1 • Standard Gamma CDF is the incomplete Gamma Function

  14. Gamma Distribution • Let X have a Gamma Distribution with parameters a and b . Then for any value x>0, the probability that Values of the Incomplete Gamma Function can be found in the Table in the back of the text.

  15. Using R to calculate Probabilities • Functions are provided to evaluate the cumulative distribution function P(X ≤ x), the probability density function. • Distribution R-name Additional Arguments • binomial binom size, prob • chi-squared chisq df, ncp • exponential exp rate • F f df1, df2, ncp • gamma gamma shape, scale • normal norm mean, sd • Student’s t t df, ncp • uniform unif min, max • Prefix the name given here by ‘d’ for the density, ‘p’ for the CDF, and `q’ for quantile (i.e., reverse lookup)

  16. Using R to calculate Probabilities: Examples • Normal distribution • “> pnorm(-1.96)”  cdf of -1.96 of X~N(0 ,1) • “>pnorm(93.6,100,5)”  cdf of 93.6 of X~N(100 ,5) • “> qnorm(.975)”  reverse lookup of .975 • Gamma distribution • “>pgamma(5,2)”  gives the cdf of a gamma distribution with an alpha value of 2 • “>qgamma(0.025,2)”  reverse lookup of a gamma distribution with an alpha value of 2 • Exponential distribution • “>pexp(5,2)”  cdf of a exponential distribution with a lambda value of 2

  17. Overview • Normal Approximation to the Binomial • Percentiles • Exponential and Gamma Distributions • Using R

  18. Homework • Practice using the Standard Normal Table • Reread 4.3, • Read 4.4

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