ISyE8843A, Brani Vidakovic Handout 0 1 Some Important Continuous Distributions 1.1 Uniform U(a; b) Distribution Random variable X has uniform U(a; b) distribution if its density is given by 8 ½ < 0; x < a 1 ; a · x · b f(xja; b) = b¡a F (xja; b) = x¡a ; a · x · b 0; else : b¡a 1; x > b k 1 bk+1¡ak+1 ² Moments: EX = b¡a k+1 ; k = 1; 2;::: (b¡a)2 ² Variance V arX = 12 : 1 eitb¡eita ² Characteristic Function '(t) = b¡a it : X¡a ² If X »U(a; b) then Y = b¡a »U(0; 1): ² Typical model: Rounding (to the nearest integer) Error is often modeled as U(¡1=2; 1=2) ² If X » F , where F is a continuous cdf, then Y = F (X) »U(0; 1): 1.2 Exponential E(¸) Distribution Random variable X has exponential E(¸) distribution if its density and cdf are given by ½ ½ ¸e¡¸x; x ¸ 0 1 ¡ e¡¸x; x ¸ 0 f(xj¸) = ;F (xj¸) = 0; else 0; else k k! ² Moments: EX = ¸k ; k = 1; 2;::: 1 ² Variance V arX = ¸2 : ¸ ² Characteristic Function '(t) = ¸¡it : ² Exponential random variable X possesses memoryless property P (X > t + sjX > s) = P (X > t): ² Typical model: Lifetime in reliability. ¡ x 0 0 1 ¸0 k 0 k ² Alternative parametrization, ¸ as scale. f(xj¸ ) = ¸0 e , EX = k!(¸ ) ; k = 1; 2;::: , V arX = (¸0)2: 1.3 Double Exponential DE(¹; σ) Distribution Random variable X has double exponential DE(¹; σ) distribution if its density and cdf are given by 1 1 ³ ´ f(xj¹; σ) = e¡jx¡¹j/σ;F (xj¹; σ) = 1 + sgn(x ¡ ¹)(1 ¡ e¡jx¡¹j/σ ; x; ¹ 2 R; σ > 0: 2σ 2 2k ¹2k ¹2(k¡1)σ2 σ2k 2k+1 2k+1 ² Moments: EX = ¹, EX = [ (2k)! + (2(k¡1))! + ::: 1 ](2k)! EX = ¹ (2k + 1)! ² Variance V arX = 2σ2: ei¹t ² Characteristic Function '(t) = 1+(σt)2 : 1 1.4 Normal (Gaussian) N (¹; σ2) Distribution Random variable X has normal N (¹; σ2) distribution with parameters ¹ 2 R (mean, center) and σ2 > 0 (variance) if its density is given by 2 1 ¡ (x¡¹) f(xj®; ¯) = p e 2σ2 ; ¡ 1 < x < 1 2¼σ2 ² Moments: EX = ¹; E(X ¡ ¹)2k¡1 = 0; E(X ¡ ¹)2k = (2k ¡ 1)(2k ¡ 3) ::: 5 ¢ 3 ¢ 1 ¢ σ2k = (2k ¡ 1)!!σ2k; k = 1; 2;::: 2 2 ² Characteristic function '(t) = ei¹t¡t σ =2: 2 X¡¹ 1 ¡ x ² Z = has standard normal distribution Á(x) = p e 2 : The cdf of standard normal distribution σ 2¼ R x is a special function ©(x) = ¡1 Á(t)dt and its values are tabulated in many introductory statistical texts. ² Standard half-Normal distribution is given by f(x) = 2Á(x)1(x ¸ 0): 2 1.5 Chi-Square Ân Distribution 2 Random variable X has chi-square Ân distribution distribution with n degrees of freedom if its density is given by ( 1 xn=2¡1e¡x=2; x > 0 f(x) = 2n=2¡(n=2) 0; else ² Moments: EXk = n(n + 2) ::: (n + 2(k ¡ 1)); k = 1; 2;:::: ² Expectation EX = n, Variance V arX = 2n, and Mode m = n ¡ 2; n > 2: ² Characteristic function '(t) = (1 ¡ 2it)¡n=2: 2 ² Noncentral chi-square ncÂn distribution is the distribution of sum of squares of n normals: N (¹i; 1): P 2 2 The non-centrality parameter is ± = i ¹i : Density of ncÂn distribution involves Bessel function of the 2 2 first kind. The simplest representation for the cdf of ncÂn is an infinite Poisson mixture of central  ’s where degrees of freedom are mixed: X1 (±=2)i F (xjn; ±) = e¡±=2P (Â2 · x): i! n+2i i=0 2 ² Ân is G(n=2; 2) or in alternative parametrization Gamma(n=2; 1=2): 2 2 2 ² Inverse Ân, inv ¡ Ân is defined as IG(n=2; 2) or IGamma(n=2; 1=2): Scaled inverse Ân, inv ¡ Â2(n; s2) is IG(n=2; 2=(ns2)) or IGamma(n=2; ns2=2): 1.6 Chi Ân Distribution Random variable X has chi-square Ân distribution distribution with n degrees of freedom if its density is given by ( 1 xn¡1e¡x2=2; x > 0 f(x) = 2n=2¡1¡(n=2) 0; else 2 k=2 n+k k 2 ¡( 2 ) ² Moments: EX = ¡(n=2) ; k = 1; 2;:::: h i2 ¡((n+1)=2) ² Variance V arX = n ¡ 2 ¡(n=2) . p 1 P1 (i 2t)k ² Characteristic function '(t) = ¡(n=2) k=0 k! ¡((n + k)=2): ² Ân for n = 2 is Raleigh Distribution, for n = 3 Maxwell distribution. 1.7 Gamma G(®; ¯) Distribution Random variable X has gamma G(®; ¯) distribution with parameters ® > 0 (shape) and ¯ > 0 (scale) if its density is given by ( 1 ®¡1 ¡x=¯ ½ ® x e ; x ¸ 0 0; x · 0 f(xj®; ¯) = ¡(®)¯ F (xj®; ¯) = 0; else inc¡(xj®; ¯); x ¸ 0 ² Moments: EXk = ®(® + 1) ::: (® + k ¡ 1)¯k; k = 1; 2;::: ² Mean EX = ®¯; Variance V arX = ®¯2: ² Mode m = (® ¡ 1)¯: 1 ² Characteristic function '(t) = (1¡it¯)® : ba a¡1 ¡bx k a(a+1):::(a+k¡1) ² Alternative parametrization, Gamma(a; b): f(xja; b) = ¡(a) x e ; x ¸ 0; EX = bk ; k = 1; 2; : : : V arX = a=b2: 2 ² Special cases E(¸) ´ G(1; 1=¸) ´ Gamma(1; ¸) and Ân ´ G(n=2; 2) ´ Gamma(n=2; 1=2): d 1 P® ² If ® is an integer, Ga(®; ¯) = ¡ ¯ i=1 log(U): If ® < 1, then if Y »Be(®; 1 ¡ ®) and Z »E(1), X = YZ »Ga(®; 1): 1.8 Inverse Gamma IG(®; ¯) Distribution Random variable X has gamma IG(®; ¯) distribution with parameters ® > 0 and ¯ > 0 if its density is given by ( 1 e¡1=(¯x); x ¸ 0 f(xj®; ¯) = ¡(®)¯®x®+1 0; else ² Moments: EXk = f(® ¡ 1)(® ¡ 2) ::: (® ¡ k)¯kg¡1; k = 1; 2;::: (® > k): 1 1 ² Mean EX = ¯(®¡1) ; ® > 1, Variance V arX = (®¡1)2(®¡2)¯2 ; ® > 2 1 ² Mode m = (®+1)¯ : ² Characteristic function '(t) = : ² If X »G(®; ¯) then X¡1 »IG(®; ¯): ba ¡b=x k bk ² Alternative parametrization, IGamma(a; b). f(xja; b) = ¡(a)xa+1 e ; x ¸ 0; EX = (a¡1):::(a¡k) ; k = 1; 2; : : : V arX = b2=((a ¡ 1)2(a ¡ 2)): 1.9 Beta Be(®; ¯) Distribution Random variable X has beta Be(®; ¯) distribution with parameters ® > 0 and ¯ > 0 if its density is given by 3 ( 8 ¡(®+¯) < 0; x · 0 x®¡1(1 ¡ x)¯¡1; 0 · x · 1 f(xj®; ¯) = ¡(®)¡(¯) F (xj®; ¯) = incBeta(xj®; ¯); 0 · x · 1 : 0; else 1; x ¸ 1 k ¡(®+k)¡(®+¯) ®(®+1):::(®+k¡1) ² Moments: EX = ¡(®)¡(®+¯+k) = (®+¯)(®+¯+1):::(®+¯+k¡1) : ® ®¯ ² Mean EX = ®+¯ ; Variance V arX = (®+¯)2(®+¯+1) : ®¡1 ² Mode m = ®+¯¡2 ; if ® > 1; ¯ > 1 ¡(®+¯) P1 (it)k ¡(®+k) ² Characteristic function '(t) = ¡(®) k=0 k! ¡(®+¯+k) ² If X1;X2;:::;Xn is a sample from the uniform U(0; 1) distribution, and X(1);X(2);:::;X(n) its order statistics. Then X(k) »Be(k; n ¡ k + 1): ² Be(1; 1) ´ U(0; 1); Be(1=2; 1=2) ´ Arcsine Law 1.10 Student tn Distribution Random variable X has Student tn distribution with n degrees of freedom if its density is given by µ ¶ n+1 n+1 2 ¡ 2 ¡( 2 ) x fn(x) = p n 1 + ; ¡ 1 < x < 1: n¼ ¡( 2 ) n k ¡(n=2¡k)¡(k+1=2) 2k pn 2k¡1 ² Moments: EX = ¼ ¡(n=2) ; 2k < n: EX = 0; k = 1; 2;:::: ² V arX = n ; n > 2 n¡2 p p ¡ njtj P ¡ ¢ p ² Characteristic function '(t) = ¼ ¡(m) e m¡1(2k)! m¡1+k (2 njtj)m¡1¡k; ¡(n=2) 22(m¡1)(m¡1)! k=0 2k n+1 for m = 2 integer. ² t2k¡1; k = 1; 2;::: is infinitely divisible. p 2 ² If Z »N (0; 1) and Y » Ân, then X = Z= Y=n has tn distribution. ² t1 ´ Ca(0; 1): 1.11 Cauchy Ca(a; b) Distribution Random variable X has Cauchy Ca(a; b) distribution with center a 2 R and scale b 2 R+ if its density is given by 1 1 b f(x) = = ; ¡ 1 < x < 1: x¡a 2 2 2 b¼ 1 + ( b ) ¼(b + (x ¡ a) ) If a = 0 and b = 1, the distribution is called standard Cauchy. The cdf is 1 x ¡ a F (x) = 1=2 + arctan ; ¡ 1 < x < 1: ¼ b ² Moments: No finite moments. Value x = a is the mode and median of the distribution. ² Characteristic function. '(t) = eiat¡bjtj: Cauchy distribution is infinitely divisible. Since '(t) = n ('(t=n)) , if X1;:::;Xn are iid Cauchy, X¹ is also a Cauchy. ² If Y; Z are independent standard normal N (0; 1) then X = Y=Z has Cauchy Ca(0; 1) distribution. If a variable U is uniformly distributed between ¡¼=2 and ¼=2, then X = tan U will follow standard Cauchy distribution. ² The Cauchy distribution is sometimes called the Lorentz distribution (especially in engineering com- munity). 4 1.12 Fisher Fm;n Distribution Random variable X has Fisher Fm;n distribution with m and n degrees of freedom if its density is given by mm=2nn=2xm=2¡1 ¡(m+n)=2 f(xjm; n) = B(m=2;n=2) (n + mx) ; x > 0: The cdf F (x) is of no closed form. ¡ ¢ k n k ¡(m=2+k)¡(n=2¡k) ² Moments E(X ) = m ¡(m=2)¡(n=2) ; 2k < n: n 2n2(m+n¡2) ² Mean EX = n¡2 ; n > 2; Variance V arX = m(n¡2)2(n¡4) ; n > 4: ² Mode n=m ¢ (m ¡ 2)=(n + 2) 1.13 Logistic Lo(¹; σ2) Distribution Random variable X has logistic Lo(¹; σ2) distribution with parameters ¹ 2 R and σ2 > 0 if its density is given by h ¡ ¢i ¼ exp ¡ p¼ x¡¹ 2 3 σ f(xj¹; σ ) = p n h ¡ ¢io2 σ 3 1 + exp ¡ p¼ x¡¹ 3 σ 2 ² Expectation: EX = ¹ Variance: V arX =p σ : p it¹ σ 3 σ 3 ² Characteristic function '(t) = e ¡(1 ¡ i ¼ t)¡(1 + i ¼ t): ² Used in modeling in biometry (drug response, toxicology, etc.) ² Alternative parametrization 1.14 Lognormal LN (¹; σ2) Distribution Random variable X has lognormal LN (¹; σ2) distribution with parameters ¹ 2 R and σ2 > 0 if its density is given by ( (ln(x)¡¹)2 1 ¡ p e 2σ2 ; x > 0 f(xj®; ¯) = xσ 2¼ 0; else k 1 2 4 ² Moments: EX = expf 2 k σ + k¹g: 4 4 ² Variance V arX = eσ +¹[eσ ¡ 1]: 2 ² Mode e¹¡σ ² If Z »N (0; 1) then X = expfσ2Z + ¹g » LN (¹; σ2).