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Chapter 8

Chapter 8. Binomial and Geometric Distributions. 8.1 The Binomial Distributions. Binomial Setting Each observation falls into one of two categories, success or failure There is a fixed number n of observations The n observations are all independent

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Chapter 8

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  1. Chapter 8 Binomial and Geometric Distributions

  2. 8.1 The Binomial Distributions • Binomial Setting • Each observation falls into one of two categories, success or failure • There is a fixed number n of observations • The n observations are all independent • The probability of success p is the same for all observations. • ALL FOUR MUST BE MET to be BINOMIAL

  3. Binomial Distribution • The distribution of the count X of successes in the binomial setting is the binomial distribution with parameters n and p. • The parameter n is the number of observations, • and p is the probability of a success on any one observation. • The possible values of X are the whole numbers from 0 to n. • B(n, p)

  4. PDF • Given a discrete random variable X, the PDF – probability distribution function – assigns a probability to each value of X • Must satisfy all probability rules • Binompdf (n, p, x) command is found under 2nd DISTR/ 0: binompdf on TI 83 • EX 8.5

  5. CDF • Given a random variable X, the CDF – cumulative distribution function – of X calculates the sum of the probabilities for 0,1,2,…, up to the value X. It calculates the probability of obtaining at most X successes in n trials.

  6. Binomial Coefficient • The number of ways of arranging k successes among n observations is given by the binomial coefficient For k = 0, 1, 2, …., n

  7. 8.3: X=type O blood, n=5, p=0.25 • B(5, 0.25) • BPDF(5, .25, 2) = 0.2637 • BPDF(n, p, # of succ) • X 0 1 2 3 4 5 L1 • Bpdf L2=binpdf(5,.25) • Bcdf L3=bincdf(5,.25) • C. Verified in CDF

  8. Histogram of PDF

  9. Histogram of CDF

  10. Binomial Probability • If X has a binomial distribution with n observations and probabilities p of success on each observation, the possible values of X are 0,1,2,….,n. If k is any one of the values

  11. #2, 5, 7, 8, 12, 13 • In class 3, 4, 6, 9, 10, 11

  12. 8.1 part 2 • Mean and Standard deviation of a binomial random variable • If a count X has a binomial distribution with number of observations n and probability of success p, the mean and standard deviation of X are • μ=np σ=√np(1-p)

  13. Normal approximation for binomial distributions • Suppose that a count X has a binomial dist. With n trials and success probability p. • When n is large, the distribution of X is approximately normal, N(np, √np(1-p)) • As a rule of thumb we will you the normal approximation when n and p satisfy • np≥10 and • n(1-p)≥10

  14. Accuracy of N(np, √np(1-p)) improves as n gets larger • Most accurate for any fixed n when p is close to ½ • Least accurate for any fixed n when p is close to 0 or 1

  15. HW#17, 19b-d, 20, 26 • In Class#15, 16

  16. 8.2:Geometric Distributions • Geometric Setting • Each observation falls into one of two categories, success or failure • The probability of success p is the same for each observation • Observations are independent • The variable of interest is the number of trials required to obtain first success

  17. Rule for calculating geometric probabilities • If X has a geometric distribution with prob p of success and (1-p) of failure on each observation, the possible values of X are 1, 2, 3, … If n is any one of these values, the prob. that the first success occurs on the nth trail is • P(X=n) = (1-p)n-1p (Formula) • GPDF(p, n) GCDF(p, n) n=1st success

  18. Mean and Standard Deviation of Geometric Random Variable • μ = 1/p mean • (1-p)/p2 variance of X • √(1-p)/p standard deviation of X

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