Question

Prove the formulas given in the table at the beginning of Section 3.4 for the Bernoulli, Poisson, Uniform, Exponential, Gamma, and Beta. Here are some hints. For the mean of the Poisson, use the fact that! ea = 2 , a"/x!. To compute the variance, first compute E(X(X - 1)). For the mean of the Gamma, it will help to multiply and divide by [(a+1)/Ba+1 and use the fact that a Gamma density integrates to 1.! For the Beta, multiply and divide by I(a + 1)(3)/T(a + 8 + 1).

Answer 1

- We know that for a random varicable x, Eet Meamz ECX), 1x PCX=x) for destrelė Sefondre for continuous. Variances ELX2) ECX) VCX ( Bernoulli Distribution- The probability mass function of a Bernoulle random variable x is given by, P(X=x) =SP: 231 (1-2, 2-0 soup<. E(X)= I x PCxox) c00x1-P) + Ixp =ļ V(x) = E(X2) - (EC*2) Ex) = 2** (42) = 0x (1-8) + x

V(X)= p- pe = pli-p 5 Poisson Distribution. 1 The prof of a Poisson random vasitle x with parameter & is given by f(x-x). Xe = - , K = 0,1,2, . . - : (+44 ) - Cl: Cr. 12 (- --*32- .:54) 1 * 4) )! X 7 * Let y=x-1 * A 8 10 12

Recall 8- Ž O yoo y -- E(X)= leden V(X= E(X2)- (EX)) E(x)= E(x(x-1) +x) - E (x(x-1)) + ECX) E(x(x10) 3262-) 78 X 2 como x=0 < 0+0+2 26-12) x=2 z 2 ** *)(x-2)) neid ei oo 2- :: EX?) Na 2. V(x) = 44 x - x x

(W) Unifoom Distöibution The probability densay function (pdf) of u(a,b) don a given by fusosada, aux et o ow Etxe ja, de forma di exse fefero de b-a , Fb-a)* (bra) Mathematical (6/a) (6+a) 2 Cbag cancells out identity bta

(2) - (ا - (۷ E(X)= 8x² I de a bra ا دعاء ها را یا وعطا) (ط) ما بطه وم = VG )= V6 ) = 1 + 3 ( 3 A3 - qa²+ hab + 46² 39 = 36 - bab 2 ا کی . های راه و شه . رسم شده - اس 12

(jv) Exponential distribution . The pdf of exponential distritution with parameter & is given by fore) $2*, *30 Ex) = Se 2 de - | Define Gamma function, B al at m7 e x daa p=(P-1)! I VOXDE E(X2)– (Ex)) Ecx)= 5 de * Sixth de ECx²). bay Yamame function, "(x)= apa pe = 2

(1) Gamma Distribution The pdf of Garrima random variable with parameters of and p is given by fon. Lehd, x70 «р 70 rapte Ex- 1 t at de - lut (Garama definition 1 + = २२ LB

V(x)= E(X)–(EX))? EGG² e P x x da sem pa de 1+2 pate lapz pop (cat) art BP Pe Tai a 4 (2+1313 V6492 (64p3 dage = 187 «pe – op?

(vi) Beta distribution. Let us recall the Beta function, B (a, b) = r at bal. Bla,b) = 1 x Cleo da -ao A randoms variable x has Beta distribution # with parameters of and &, if its pdf is given by, a a-l By fos I 2 (1-10" ,oses) BCdoB) dx 1. (-x) dx Ecx)= 1x I x (1-x) o BG, B) 2+1-1 3-4 Bl2,53 a t B(&H, B) Bla,B) 1. Cat RH BB - Tap Tetpt (Bla,b)= falb ſatb)

o B6,8) 1 2+2-| B-1, N(X)= F(x2)–(Ecx)) - E(X²)= (x² - Ret! ( 15 de - * o-n be EI B(4+2, BD B6,B] 1637 Patz B - Ta B Titpt2 - Tatß Pati (&+) | a d+P(4+ Ph!) = PAB Qx & (641) R4р (ар) С<tp+) e L(441) Auß) (4+3+1) U42= 2 (2+12 -2² Simplifying (+B)(4+5+1 6+p3 = 2 (2+1) (2+8) dla +8+1) (2+8) ² (2+8+1) Q+8)(2+8+1)

(i+d+x)/x-(4+x)(1+0) įretje) (It&ty = (2+2)(4b)-d2_28-02 @tppe Ca4b+1) = 8 + 88 +² +28 - 28-48- &+3² (2+8+1) 2 = 2B Expo" (248+1)