1 / 18

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

mitch
Download Presentation

Chapter 8

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related