Notes on Binomial Distributions - Probability Statistics Bioinformatics | MATH 186, Study notes of Mathematics

Download Notes on Binomial Distributions - Probability Statistics Bioinformatics | MATH 186 and more Study notes Mathematics in PDF only on Docsity! Math 186, Winter 2005, Prof. Tesler – January 21, 2005 Binomial distribution Here are the tabulated values and graphs of the discrete probability density function (pdf) and cumulative distribution function (cdf) for the binomial distribution with parameters n = 10 and p = .75. pX(k) = P (X = k) = { ( 10 k ) (.75)k(.25)n−k if k = 0, 1, . . . , 10; 0 otherwise. FX(k) = P (X ≤ k) =   0 if k < 0;∑bkc r=0 ( 10 r ) (.75)r(.25)n−r if 0 ≤ k ≤ 10; 1 if k ≥ 10. Note: bxc is the “floor” function (greatest integer ≤ x), which you may have seen written [x] elsewhere: any real number x can be written uniquely as x = m + δ, where m is an integer and δ is a real number with 0 ≤ δ < 1, and the floor of x is defined as bxc = m. For example, b3c = 3, b−3c = −3, b3.2c = 3, b−3.2c = −4. pdf cdf k pX(k) FX(k) k < 0 0 0 0.00000095 0 ≤ k < 1 0.00000095 1 0.00002861 1 ≤ k < 2 0.00002956 2 0.00038624 2 ≤ k < 3 0.00041580 3 0.00308990 3 ≤ k < 4 0.00350571 4 0.01622200 4 ≤ k < 5 0.01972771 5 0.05839920 5 ≤ k < 6 0.07812691 6 0.14599800 6 ≤ k < 7 0.22412491 7 0.25028229 7 ≤ k < 8 0.47440720 8 0.28156757 8 ≤ k < 9 0.75597477 9 0.18771172 9 ≤ k < 10 0.94368649 10 0.05631351 10 ≤ k 1.00000000 other 0 0 5 10 0 0.2 0.4 0.6 0.8 1 k p X (k ) Discrete probability density function 0 5 10 0 0.2 0.4 0.6 0.8 1 k F X (k ) Cumulative distribution function Sample uses of tables: P (X ≤ −3.2) = 0 P (X ≤ 12.8) = 1 P (X ≤ 6.5) = FX(6.5) = 0.22412491 P (X = 6.5) = pX(6.5) = 0 P (X ≤ 6) = FX(6) = 0.22412491 P (X = 6) = pX(6) = 0.14599800 P (X < 6) = FX(6−) = 0.07812691 (Convert P (X < a) into “P (X ≤ a−)” = FX(a−)) P (X > 6) = 1 − P (X ≤ 6) = 1 − FX(6) = 1 − 0.22412491 = 0.77587508 P (4 < X ≤ 8) = P (X ≤ 8) − P (X ≤ 4) (Note: X ≤ 4 is contained in the event X ≤ 8) = FX(8) − FX(4) = 0.75597477 − 0.01972771 = 0.55869767 P (4 ≤ X ≤ 8) = “P (4− < X ≤ 8)” = FX(8) − FX(4−) = 0.75597477 − 0.00350571 = 0.75246906 P (4 < X < 8) = “P (4 < X ≤ 8−)” = FX(8−) − FX(4) = .47440720 − 0.01972771 = .45467949 P (4 ≤ X < 8) = “P (4− < X ≤ 8−)” = FX(8−) − FX(4−) = 0.47440720 − 0.00350571 = 0.47090149 An alternate way to compute these is to take advantage of the discrete values being integers; instead of using “a−” we can go down to a − 1: P (X < 6) = P (X ≤ 5) = FX(5) = 0.07812691.