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1 M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term Clculte the men, vrince nd chrcteristic function of the following probbility density functions. ) The exponentil distribution with density with λ >. fx) b) The uniform distribution with density with < b. fx) { λe λx x >, x <, { 1 b < x < b, x /, b), c) The Gmm distribution with density { λ fx) Γα) λx)α 1 e λx x >, x <, with λ >, α > nd Γα) is the Gmm function SOLUTION ) EX) Γα) 1 λ. + ξ α 1 e ξ dξ, α >. xfx) dx λ + xe λx dx EX 2 ) + 2 λ 2. x 2 fx) dx λ + x 2 e λx dx 1

2 b) Consequently, The chrcteristic function is vrx) EX 2 ) EX) 2 1 λ 2. φt) Ee itx ) λ EX) + + b 2. e itx e λx dt xfx) dx b λ λ it. x b dx c) Consequently, EX 2 ) The chrcteristic function is + x 2 fx) dx λ b2 + b vrx) EX 2 ) EX) 2 φt) Ee itx ) λ b EX) λα Γα) λ b x 2 x2 b )2. 12 b dx e itx 1 b dt eitb e it itb ). + Γα + 1) λγα) α λ. x α e λx dx Consequently, The chrcteristic function is EX 2 ) λ + Γα + 2) λ 2 Γα) x 1+α e λx dx αα + 1) λ 2. vrx) EX 2 ) EX) 2 α λ 2. φt) Ee itx ) λα Γα) λα 1 Γα) λ it) α λ α λ it) α. e itx x α 1 dt e y y α 1 dy 2

3 2. ) Let X be continuous rndom vrible with chrcteristic function φt). Show tht EX k 1 i k φk) ), where φ k) t) denotes the k-th derivtive of φ evluted t t. b) Let X be nonnegtive rndom vrible with distribution function F x). Show tht EX) + 1 F x)) dx. c) Let X be continuous rndom vrible with probbility density function fx) nd chrcteristic function φt). Find the probbility density nd chrcteristic function of the rndom vrible Y X + b with, b R. d) Let X be rndom vrible with uniform distribution on [, 2π]. Find the probbility density of the rndom vrible Y sinx). SOLUTION ) We hve Consequently Thus: φt) Ee itx ) R R e itx fx) dx. φ k) t) ix) k e itx fx) dx. φ k) ) ix) k fx) dx i k EX k, R nd EX k 1 i k φ k) ). b) Let R > nd consider Thus, PX < R) R R xfx) dx x df dx dx xf x) R R R F x) dx F R) F x)) dx. EX lim 3 R PX < R) 1 F x)) dx,

4 where the fct lim x F x) 1 ws used. Alterntively: 1 F x)) dx x y fy) dydx fy) dxdy yfy) dx EX. c) We hve: PY y) PX + b y) PX y b ) y b fx) dx. Consequently, Similrly, f Y y) y PY y) 1 ) y b f. d) The density of the rndom vrible X is φ Y t) Ee ity Ee itx+b) e itb Ee itx e itb φt). f X x) { 1 2π, x [, 2π],, x / [, 2π]. The distribution function is F X x) { x <, x 2π, x [, 2π], 1, x > 2π. The rndom vrible Y tkes vlues on [ 1, 1]. Hence, PY PY y) 1 for y 1. Let now y 1, 1). We hve y) for y 1 nd F Y y) PY y) PsinX) y). The eqution sinx) y hs two solutions in the intervl [, 2π]: x rcsiny), π rcsiny) for y > nd x π rcsiny), 2π + rcsiny) for y <. Hence, F Y y) π + 2 rcsiny), y 1, 1). 2π 4

5 The distribution function of Y is F Y y) { y, π+2 rcsiny) 2π, y 1, 1), 1, y 1. We differentite the bove expression to obtin the probbility density: f Y y) { 1 π 1, 1 y 2 y 1, 1),, y / 1, 1). 3. Let X be discrete rndom vrible tking vles on the set of nonnegtive integers with probbility mss function p k PX k) with p k, + k p k 1. The generting function is defined s ) Show tht gs) Es X ) where the prime denotes differentition. k p k s k. EX g 1) nd EX 2 g 1) + g 1), b) Clculte the generting function of the Poisson rndom vrible with p k PX k) e λ λ k, k, 1, 2,... nd λ >. k! c) Prove tht the generting function of sum of independent nonnegtive integer vlued rndom vribles is the product of their generting functions. ) We hve Hence nd from which it follows b) We clculte g s) k g 1) kp k s k 1 nd g s) k g 1) k 2 p k k k k kp k EX EX 2 g 1) + g 1). gs) 5 k kk 1)p k s k 2. kp k EX 2 g 1) e λ λ k k! e λs 1). s k

6 c) Consider the independent nonnegtive integer vlued rndom vribles X i, i 1,... d. Since the X i s re independent, so re the rndom vribles e X i, i 1,.... Consequently, g P d i1 X s) d EeP i1 X i ) Π d i i1ee X i ) Π d i1g Xi s). 4. Let b R n nd Σ R n n symmetric nd positive definite mtrix. Let X be the multivrite Gussin rndom vrible with probbility density function 1 1 γx) 2π) n/2 exp 1 ) detσ) 2 Σ 1 x b), x b. ) Show tht R d γx) dx 1. b) Clculte the men nd the covrince mtrix of X. c) Clculte the chrcteristic function of X. ) From the spectrl theorem for symmetric positive definite mtrices we hve tht there exists digonl mtrix Λ with positive entries nd n orthogonl mtrix B such tht Let z x b nd y Bz. We hve Σ 1 B T ΛB. Σ 1 z, z B T ΛBz, z ΛBz, Bz Λy, y d λ i yi 2. Furthermore, we hve tht detσ 1 ) Π d i1 λ i, tht detσ) Π d i1 λ 1 i of n orthogonl trnsformtion is J detb) 1. Hence, R d exp 1 ) 2 Σ 1 x b), x b i1 dx exp R d exp 1 R d 2 Π n i1 exp R 1 ) 2 Σ 1 z, z dz nd tht the Jcobin ) d λ i yi 2 J dy i1 1 2 λ iy 2 i 2π) n/2 Π n i1λ 1/2 i ) dy i 2π) n/2 detσ), from which we get tht R d γx) dx 1. 6

7 b) From the bove clcultion we hve tht Consequently γx) dx γb T y + b) dy 1 2π) n/2 detσ) Πn i1 exp 1 ) 2 λ iyi 2 dy i. EX xγx) dx R d B T y + b)γb T y + b) dy R d b γb T y + b) dy b. R d We note tht, since Σ 1 B T ΛB, we hve tht Σ B T Λ 1 B. Furthermore, z B T y. We clculte EX i b i )X j b j )) z i z j γz + b) dz R d ) 1 2π) n/2 B ki y k B mi y m exp 1 λ l yl 2 dy detσ) R d 2 k m l ) 1 2π) n/2 B ki B mj y k y m exp detσ) R 1 λ l yl 2 dy d 2 k,m B ki B mj λ 1 k δ km k,m Σ ij. c) Let y be multivrite Gussin rndom vrible with men nd covrince I. Let lso C B Λ. We hve tht Σ CC T C T C. We hve tht X CY + b. To see this, we first note tht X is Gussin since it is given through liner trnsformtion of Gussin rndom vrible. Furthermore, EX b nd EX i b i )X j b j )) Σ ij. l Now we hve: φt) Ee i X,t e i b,t Ee i CY,t e i b,t Ee i Y,CT t e i b,t Ee i P j P k C jkt k )y j e i b,t e 1 2 Pj P k C jkt k 2 e i b,t e 1 2 Ct,Ct e i b,t e 1 2 t,ct Ct e i b,t e 1 2 t,σt. 7

8 Consequently, φt) e i b,t 1 2 t,σt. 8

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