Special Discrete RV s. Then X = the number of successes is a binomial RV. X ~ Bin(n,p).

Transcription

1 Sect 3.4: Binomial RV Special Discrete RV s 1. Assumptions and definition i. Experiment consists of n repeated trials ii. iii. iv. There are only two possible outcomes on each trial: success (S) or failure (F). Trials are independent Prob. of success is p on each trial. Then = the number of successes is a binomial RV. ~ Bin(n,p). 1

2 . pmf p( x) for x = 0, 1,, n; 0 else n b( x; n, p) p x (1 p) x nx 3. Facts If ~ Bin(n, p) and Y = number of failures, then Y ~ Bin(n, 1- p) np and np(1 p)

3 Sect. 3.5: Hypergeometric RV 1. Assumptions and definition: N objects of types, success or failure. Assume M successes (N-M failures). Random sample of n objects, without replacement (means trials are dependent). = number of successes is a hypergeo. RV.. pmf M N M x n x p ( x) h(x ; n, M, N) N n for x = 0, 1,, n ; x < M ; and n - x < N M; 0 else 3

4 3. Facts n M N 4. Calculations: Calc or comp., but if N is large and n is small compared to N, approximate hypergeo. prob.s by binomial probs: When: text (p. 109) says n <.05 N M M N n n 1 N N N 1 Bin( n, p M / N) Remark: If n is also large, we may approximate this binomial by a normal or a Poisson 4

5 Sect 3.5 Negative Binomial RV 1. Assumptions and definition: Experiment consists of repeated trials such that i. Trials are independent. ii. Only two possible outcomes on each trial. success or failure, 1 or 0, etc. iii. Prob. of success is p on each trial. Let = the number of failures until we get the r th success. Then is a neg. bin. RV: ~ NBin(r, p). Key: similar to binomial, but here the number of successes is fixed, the number of trials is random 5

6 . pmf x r 1 p( x) nb( x; r, p) p r (1 r 1 for x = 0, 1,. p) x 3. Facts r(1 p) r ( 1 p) p p

7 Sect 3.6: Poisson Distribution 1. Assumptions and definition: is the count of the number of occurrences of occurrences during a fixed time period i. Occurrences are independent. ii. Occurrences happen one-at-time. iii. Occur at a fixed, constant rate. Then is a Poisson RV: ~ Poi(l) where l > 0 is the average number of occurrences in the time period.. pmf x l l e p( x; l) for x x! 0,1, 7

8 3. Facts: l l 4. Finding probs: calculator/computer. Notes Tricky part: getting l right. If a = rate = # occurrences per unit time and t = length of time interval (in the same time units), then l a t Approximating binomial probs: Suppose ~ Bin(n, p) with n large, p is very small. Then Poi( l np) o When? Text: n > 50, and np < 5. o If p is very large: switch the role of success and failure. 8

9 Discrete RV Counts in S/F experiments Binomial Bin(n, p) Negative binomial NBin(r, p). Hypergeometric n, N, M, Trials: n fixed; indep. Random; indep. n fixed; depend. RV = # S s = # fail s till r th S = # S s Approx n large, np > 10, n(1- p) > 10, use: N(np, np(1-p) ) N large; n <.05 N, use: Bin(n, p = M/N). n large, p small, use: Poi(l = np) (p large, change S & F) Poisson: Poi(l). Count occurrences in time (or space); l = average in time interval 9

10 Chap 4: Continuous RV Continuous RV can take on any number an interval. (Not isolated values like discrete case) Probability density function (pdf) f(x) of a continuous RV has two key properties: 1. f(x) > 0 for all x.. Prob. is in A = integral of pdf over A: P ( A) f ( x) dx Notes: We must have pdf is not a prob. for a continuous RV, P( = x) = 0 A f ( x) dx 1 i.e., Prob = area under f 10

11 Cumul. Dist. Functions (cdf): F(x) = P( < x) 0 < F(x) < 1, for all x As, F As x ( x) 0 x, F( x) 1 F(x) is non-decreasing Regions where F is flat are regions of prob. = 0 For a continuous RV, F(x) is a continuous function of x Wherever the derivative exists, Same as discrete RV f ( x) df( x) dx

12 Expected Values Similar ideas as discrete case: Let be continuous RV with pdf f. The expected value of or expectation of or mean of, is if finite General: the expected value of h() is if finite. E[ ] xf ( x) dx E [ h( )] h( x) f ( x) dx 1

13 Useful Facts and Definitions For constants a and b, E[ a b ] a be( ) E [ ah ( ) bh ( )] ae[ h ( )] be[ h ( 1 1 )] constants come out of E s and E(sum) = sum(e s) 13

14 Measures of Spread of a Prob. Dist Variance of is Standard deviation (SD) of is the square-root of the variance Facts V( ) V( ) E( ) E( ) [ E( V( a b) a E[( ) )] ] SD( a b) a 14

15 Percentiles (p. 139) The (100 p) th percentile of the dist. of a RV is a number h(p) such that That is, F(h(p)) = p P( < h(p)) = p Ex) the 50 th percentile is the median of the dist. 15

16 Sect 4.3: Normal Distribution 1. Many applications!!. pdf: pdf is symmetric about : mean = median = mode Finding prob s: f 1 ( x;, ) exp Step 1: Tabulate probabilities for standard normal distribution: Z ~ N(0, 1) Step : Standardization. ~ N( ) ( x ) ), 16

17 Table A.3 The cdf of Z is denoted F instead of F): P(Z < z) = F(z).

18 18 Step : Standardization Thm: Hence, That is, we can transform an -problem into a Z-problem and then use the Table to find prob s (0,1) ~ ), ( ~ N N F b b Z P b P ) (

19 Normal Approx. to the Binomial Suppose n is large and np > 10 and n(1-p) > 10 For ~ Bin(n, p), P( x) F x.5 np np(1 Recall: if n is large, but p is too small (or too large) to permit normal approx., try the Poisson approx to binomial p) 19