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 – The probability of success p is the same for all observations. • ALL FOUR MUST BE MET to be BINOMIAL 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) 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 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. 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 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 Bpdf Bcdf C. Verified in CDF 4 5 L1 L2=binpdf(5,.25) L3=bincdf(5,.25) Histogram of PDF Histogram of CDF 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 • #2, 5, 7, 8, 12, 13 • In class 3, 4, 6, 9, 10, 11 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) 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 • 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 • HW#17, 19b-d, 20, 26 • In Class#15, 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 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 Mean and Standard Deviation of Geometric Random Variable • μ = 1/p • (1-p)/p2 • √(1-p)/p mean variance of X standard deviation of X