Distribution ------ Summaries 476 Distribution Summaries Discrete name P(k) = P(X = k) mean vanance and range for k E range uniform 1 a+b (b-a+1)2-1 -- on {a, a + L, , , ,b} b-a+1 2 12 Bernoulli (p) P(l) =p: P(O) = 1- p p p(l- p) on {a, I} binomial (n., p) (npk(l - p),,-k np np(l - p) on {O, L ' . , , n} Poisson (11) ,,-11 , 1," -- /1 on {a, L 2, ... } h' 11 (C) (~- C) hypergeometric N, k n _. k nC (n. C) - II (C) CV~C) (N-n) on{O ....• n} C~) N N 1'v N-1 geometric (p) 1 1-p (1 __ p)k-Ip - -- on {L2,:L.} , P p2 geometric (p) 1-p 1-p (l-p)"p -- -- on {O.l, 2".} P p2 negative binomial (r. p) (k+r-l)p'(l_P)" r(l - p) r(l-p) on {a, 1,2, ... } r - 1 P p2 I Distribution Summaries 477 Continuous t uudefincd. density f (.r) c.d.±". F(:r) name range .\Ican Variancc for .r E range for :r E range 1 .r ~ 0 o+iJ (Ii ~ (/)2 uniform (a. Ii) (0. iJ) -- -- -- b~a 1i~1I. 2 12 1 --.-1 r 2 normal (0.1) (~x.x) --e 2' cp(.t) 0 1 V27T 1 -- I I _1.(:r_I,)2/()"2 2 nonna] (fl.. ( 2 ) (~:x.:x) --e 2 cP 1/ a V27Ta C: !l) I I I I I exponential (A) I (0. x) A,,-A.l" 1 ~ ( AJ' I/A 1/>.2 = gamma (l. A) I I I I --l .' 1 (AX)' j 1~e-AIL-- I ganllna (r. A) (O.x) r (r) -I >.':r'- 1e- A., k' r / A r/A2 I I !-c=o I i for integer r I I I chi-square (1/) as above for A = ~. I (O.x) r( ¥)-1 (~)'i -1 c· 2 1/ 211 = gamma (¥. ~ ) I' = ¥ if II is e\"pn I I I I I I I - I ! 1 /iT 1 ~ 7f r ~ r- -I Ra~']('igh (O,x) ,ce-:z 1 -- ! V"2 2 I I J I see Exercise cJ.6.5 r I beta (1',8) (0. 1) rr,. + 8) X,-l (1 ~ rj'-1 -- \'(r)[(s) . for integer,. and "' r+s [ I I (, +'1';") ;;-+ II . I I I I I 1 1 I I I 1 1 I I Callch~' (~x,x) - + ~ arrtan(.l') t 7f(l :r2) I + 2 7f I I ! 1 f I .. I ! ~ arcsine I I 1 2 I I (0.1) ~ arcsin( If) - I - =beta (1/2,1/2) 7fvx(1 ~ :r) 7f 2 11 I I I I I I i I 478 Distribution Summaries Beta Parameters: r > 0 and s > 0 Range: x E [0,1] Density function: where 1 B(r, s) = r xr-1(1 _ x)S-ldx = r(r)f(s) Jo r(r+s) is the beta function, and f(r) is the gamma function (see gamma distributions). Cumulative distribution function: (Exercises in Section 4.6.) No simple general formula for r or s not an integer. See tables of the incomplete beta function. For integers rand s 1'+8-1 ( ) S 1 i P(Xr,s ::; x) = L r +: -1 xi(l _ xr+ - - (0::; x ::; 1) z=r Mean and standard deviation: (4.6) r E(Xrs) =- , r+s Special cases: • r = s = 1: The uniform [0, 1] distribution. • r = s = 1/2: The arcsine distribution. Sources and applications: • Order statistics of uniform variables (4.6). • Ratios of gamma variables (5.4). • Bayesian inference for unknown probabilities. Normal approximation: • Good for large rand s. Distribution Summaries 479 Binomial Parameters: n = number of trials (n = 1,2, ... ) p = probability of success on each trial (0 :s p ~ 1) Range: k E {O, L ... ,n} Probability function: (2.1) P(k) = P(5 = k) = G)pk(1 - p)n~k (k = 0,1, ... , n) 5 (number of successes in n independent trials with) where = probability p of success on each trial = Xl + ... + Xn where Xi = indicator of success on trial i. Mean and standard deviation: (3.2, 3.3) E(5) = IL = np 5D(5) = a = VrLp(1 - p) Mode: (2.1) int(np+p) Consecutive odds ratios: (2.1) P(k) (n-k+l) p (decreasing) P(k - 1) k I-p Special case: 0.3) Binomial (1. p) == Bernoulli (p), distribution of the indicator of an event A with probability P(A) = p. Normal approximation: (2.2, 2.3) If (J = vInP( 1 - p) is sufficiently large P(k) ~;¢1 . (k-a~ - IL) where ¢( z) is the standard normal density function where <l> is the standard normal cumulative distribution function. Poisson approximation: (2.4) If P is close to zero P(k) ~ e~l"lLk /k! where 11 = np 480 Distribution Summaries Exponential Parameter: A > 0, the rate of an exponential random variable T. Range: t E [0,00) Density function: (4.2) P(T E dt)/dt = Ae-)"t (t ~ 0) Cumulative distribution function: (4.2) P(T -S t) = 1 - e-)"t (t ~ 0) Often T is interpreted as a lifetime. Survival function: P(T > t) = e-)"t (t:::: 0) Mean and Standard Deviation: (4.2) E(T) = l/A SD(T) = l/A Interpretation of A: A = P(T E dtlT > t)/dt is the constant hazard rate or chance per unit time of death given survival to time t. See Section 4.3 for a discussion of non-constant hazard rates. Characterizations: • Only distribution with constant hazard rate. • Only distribution with the memoryless property P(T > t + siT> t) = P(T > s) for all s, t > 0 Sources: • Time until the next arrival in a Poisson process with rate A. • Approximation to geometric (p) distribution for small p. • Approximation to beta (1, s) distribution for large s. • Spacings and shortest spacings of uniform order statistics. Distribution Summaries 481 Gamma Parameters: r > 0 (shape) A > 0 (rate or inverse scale) Range: t E [0, CX)) Density function: (4.2,5.4) P(Tr,A E dt)/dt = f(r)~l Artr~le~At (t ~ 0) where f(r) = Jooo tr~le~tdt is the gamma function. Note: f(r) = (r - 1)! for integer r. Cumulative distribution function: (4.2) No formula for non-integer r. See tables of the incomplete gamma function. For integer l' where Nt,A denotes the number of points up to time t in a Poisson process with rate A, and has Poisson (At) distribution. Mean and standard deviation: (4.2) Special cases: • gamma (1, A) is exponential (A). • gamma (71/2,1/2) is chi-square (n), the distribution of the sum of the squares of n independent standard normals. Sources: • Sum of l' independent exponential (A) variables. • Time until the nh arrival in a Poisson process with rate A. • Bayesian inference for unknown Poisson rates. • Approximation to negative binomial (1', p) for small p. • Approximation to beta (1', s) for large s. Transformations: (Notation: X rv F means X is a random variable with distribution F.) Scaling: T rv gamma (1', A) {==} AT rv gamma (1',1) Sums: For independent Ti rv gamma (ri' A) Li Ti rv gamma (Li ri, A) Ratios: For independent Tr,A and T."A TrA T~ rv beta(r, s) independent of the sum Tr,A + T."A rv gamma (1' + 8, A) r,A + S,A Higher moments: For s > 0 E[(T. )S] = f(r + s) r,A r( r )A8 Normal approximation: If l' is sufficiently large, the distribution of the standardized gamma variable Zr·A = [Tr,A - E(Tr,A)]/ SD(Tr,A) is approximately standard normal. 482 Distribution Summaries Geometric and Negative Binomial Geometric Parameter: p = success probability. Range: n E {1, 2, ... } DefInition: Distribution of the waiting time T to first success in independent trials with prob­ ability p of success on each trial. Probability function: 0.6, 3.4) P(n) = P(T = n) = (1 - p)n-lp (n = 1,2, ... ) Let F = T - 1 denote the number of failures before the first success. The distribution of F is the geometric distribution on {O, 1,2, ... }. Tail probabilities: P(T > n) = P(first n trials are failures) = (1 _ p)n Mean and Standard Deviation: (3.4) E(T) = l/p SD(T)=~/p Negative Binomial Parameters: p = success probability, r = number of successes. Range: n E {O, 1,2, ... } DefInition: Distribution of the number of failures Fr before the rth success in Bernoulli trials with probability p of success on each trial. Probability function: (3.4) P(Fr=n)=P(Tr=n+r)= (n+r-1)pr(1_p)n (n=O,l, ... ) r - 1 where Tr is the waiting time to the rth success. The distribution of Tr = Fr + r is the negative binomial distribution on {r, r + 1, ... }. Mean and standard deviation: (3.4) E(Fr) = r(l- p)/p SD(Fr) = Jr(l - p)/p Sum of geometries: The sum of r independent geometric (p) random variables on {O, 1, 2, ... } has negative binomial (r. p) distribution. Distribution Summaries 483 Hypergeometric n== sample size Parameters: N = total population size G = number of good elements in population Range: 9 E {a, l. ... , 71} Deitnition: The hypergeometric (71, N, G) distribution is the distribution of the number 8 of good elements in a random sample of size 71 without replacement from a population of size N with G good elements and B = N - G bad ones. Probability function: (2.5) P( ) = P(8 = ) = ((n) (G)g(Bh = (~) (~) 9 9 9 (N)n (~) is the chance of getting 9 good elements and b bad elements in the random sample of size 71.