ECON 240A Power 5 1 Last Tuesday Lab Two Probability Discrete Binomial Probability Distribution 2 The Normal Distribution 3 Outline The cumulative distribution function or sigmoid S shaped curve The normal distribution as an approximation to the binomial The standardized normal variable z sample means The distribution of the sample mean the normal distribution 4 Binomial Distribution Five Flips of a Fair Coin density cumulative k n p 0 5 0 03125 0 03125 1 5 0 15625 0 1875 2 5 0 3125 0 5 3 5 0 3125 0 8125 4 5 0 15625 0 96875 5 5 0 03125 1 Probability Density Function Five Flips of a Fair Coin 0 35 0 3 Probability 0 25 0 2 0 15 0 1 0 05 0 0 1 2 3 Number of Heads 4 5 6 Cumulative Distribution Function Five Flips of a Fair Coin 1 0 9 0 8 Probability 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 1 2 3 Number of Heads 4 5 7 Cumulative Distribution Function The probability of getting two or less heads in five flips is 0 5 can use the cumulative distribution function can use the probability density function and add the probabilities for 0 1 and 2 heads the probability of getting two heads or three heads is can add the probabilities for 2 heads and three heads from the probability density function 8 Cumulative Distribution Function Five Flips of a Fair Coin 1 0 9 0 8 Probability 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 1 2 3 Number of Heads 4 5 9 Binomial Distribution Five Flips of a Fair Coin density cumulative k n p 0 5 0 03125 0 03125 1 5 0 15625 0 1875 2 5 0 3125 0 5 3 5 0 3125 0 8125 4 5 0 15625 0 96875 5 5 0 03125 1 Cumulative Distribution Function the probability of getting two heads or three heads is can add the probabilities for 2 heads and three heads from the probability density function can use the probability of getting up to 3 heads P 3 or less heads from the cumulative distribution function CDF and subtract the probability of getting up to one head P 1 or less heads 11 Probability Density Function Five Flips of a Fair Coin 0 35 0 3 Probability 0 25 0 2 0 15 0 1 0 05 0 0 1 2 3 Number of Heads 4 5 12 Binomial Distribution Five Flips of a Fair Coin density cumulative k n p 0 5 0 03125 0 03125 1 5 0 15625 0 1875 2 5 0 3125 0 5 3 5 0 3125 0 8125 4 5 0 15625 0 96875 5 5 0 03125 1 Binomial Distribution Five Flips of a Fair Coin density cumulative k n p 0 5 0 03125 0 03125 1 5 0 15625 0 1875 2 5 0 3125 0 5 3 5 0 3125 0 8125 4 5 0 15625 0 96875 5 5 0 03125 1 Cumulative Distribution Function Five Flips of a Fair Coin 1 p h 3 0 9 0 8 Probability 0 7 p h 1 0 6 0 5 0 4 0 3 0 2 0 1 0 0 1 2 3 Number of Heads 4 5 15 For the Binomial Distribution Can use a computer as we did in Lab Two Can use Tables for the cumulative distribution function of the binomial such as Table 1 in the text in Appendix B p B 1 need a table for each p and n 16 17 Normal Approximation to the binomial Fortunately for large samples we can approximate the binomial with the normal distribution as we saw in Lab Two 18 Binomial Probability Density Function Forty Tosses of a Fair Coin 0 14 0 12 0 08 0 06 0 04 0 02 Number of Heads 39 36 33 30 27 24 21 18 15 12 9 6 3 0 0 Probabilty 0 1 Binomial Cumulative Distribution Function Forty Tosses of a Fair Coin 1 0 9 0 8 Probability 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 3 6 9 12 15 18 21 24 27 Number of Heads 30 33 36 39 The Normal Distribution What would the normal density function look like if it had the same expected value and the same variance as this binomial distribution from Power 4 E h n p 40 1 2 20 from Power 4 VAR h n p 1 p 40 1 2 1 2 10 21 Normal Density Function Mean 20 Variance 10 0 14 0 12 Density 0 1 0 08 0 06 0 04 0 02 0 0 10 20 30 Number of Heads 40 50 Comparing the Normal Density with the Binomial Probability Distribution 0 14 0 12 binomial normal 0 08 0 06 0 04 0 02 Number of Heads 39 36 33 30 27 24 21 18 15 12 9 6 3 0 0 Density 0 1 Comparing the Binomial and Normal Distribution Functions 1 0 9 0 8 binomial normal 0 6 0 5 0 4 0 3 0 2 0 1 Number of Heads 39 36 33 30 27 24 21 18 15 12 9 6 3 0 0 Probability 0 7 Comparing the Binomial and Normal Mean 19 5 Cumulative Distribution Functions 1 2 binomial normal 0 8 0 6 0 4 0 2 39 36 33 Number of Heads 30 27 24 21 18 15 12 9 6 3 0 0 Probability 1 Normal Approximation to the Binomial De Moivre P a k b P a n p z b n p np 1 p np 1 p This is the probability that the number of heads will fall in the interval a through b as determined by the normal cumulative distribution function using a mean of n p and a standard deviation equal to the square root of n p 1 p i e the square root of the variance of the binomial distribution The parameter 1 2 is a continuity correction since we are approximating a discrete function with a continuous one and was the motivation of using mean 19 5 instead of mean 20 in the previous slide Visually this seemed to be a better approximation than using a mean of 20 Guidelines for using the normal approximation n p 5 n 1 p 5 27 The Standardized Normal Variate Z N 0 1 0 E z 0 VAR Z 1 28 f z 1 2 e 1 2 z 0 1 2 Density Function for the Standardized Normal Variate 0 45 0 4 0 35 Density 0 3 0 25 0 2 0 15 0 1 0 05 5 4 3 2 1 0 0 1 Standard Deviations 2 3 4 5 Cumulative Distribution Function for a Standardized Normal Variate 1 0 9 0 8 Probabilty 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 5 4 …
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