Moments and Absolute Moments of the Normal Distribution

1 Moments and Abso lute Moments of the Normal Distrib ution Andreas W inkelbauer Institute of T elecommunica tions, V ienna Univ ersity of T echnology Gussha usstrasse 25/389, 1040 V ienna, Aus tria email: andreas.winkelbaue r@nt.tuwien.ac.at Abstract W e p resent formulas for the (raw and central) momen ts and absolute moments of the normal distribution. W e note that these results are not new , yet man y textbook s miss ou t on at least some of them. Hence, we belie ve that it is worthwhile to collect th ese formu las and their deriv ations in these notes. I . I N T R O D U C T I O N Let X ∼ N ( µ, σ 2 ) b e a normal (Gau ssian) rando m variable (R V) with mean µ = E { X } and variance σ 2 = E { X 2 } − µ 2 (here, E {·} d enotes expectation). In what follows, we giv e form ulas and deriv ations for E  X ν  , E  ( X − µ ) ν  , E  | X | ν  , and E  | X − µ | ν  , i.e., for the (raw) moments, the central moments, the (raw) ab solute moments, and the central absolute mo ments. W e note that the f ormulas we present hold for r eal-valued ν > − 1 . The rem ainder of this text is structure d as follows: Section I I deals with preliminar ies and introduce s notation, particularly regarding some special fun ctions. In Section I II we pr esent the results; the corr espondin g deri vations are gi ven in Section IV. I I . P R E L I M I NA R I E S W e d enote the standard deviation by σ = √ σ 2 . The imagin ary un it is j = √ − 1 and z ∗ denotes the complex conjuga te of z . Th e nonn egati ve integers are d enoted by N 0 = N ∪ { 0 } . Next, we g i ve the d efinitions of subseq uently used special functions (cf. [1]). • Gamma function: Γ( z ) , Z ∞ 0 t z − 1 e − t dt. (1) • Rising factorial: z n , Γ( z + n ) Γ( z ) (2) = z ( z + 1) · · · ( z + n − 1) , n ∈ N 0 . (3) 2 • Double factorial: z !! , r 2 z +1 π Γ  z 2 + 1  (4) = z · ( z − 2) · . . . · 3 · 1 , z ∈ N od d . (5) • K ummer’ s confluen t hyp er geometric fu nctions: Φ( α, γ ; z ) , M ( α, γ , z ) = 1 F 1 ( α ; γ ; z ) = ∞ X n =0 α n γ n z n n ! . (6) • T ricomi’ s confluent h yper geometric functions: Ψ( α, γ ; z ) , U ( α, γ , z ) = Γ(1 − γ ) Γ( α − γ + 1) Φ( α, γ ; z ) + Γ( γ − 1) Γ( α ) z 1 − γ Φ( α − γ + 1 , 2 − γ ; z ) . (7) • P arabolic cylinder functions: D ν ( z ) , 2 ν / 2 e − z 2 / 4 " √ π Γ  1 − ν 2  Φ  − ν 2 , 1 2 ; z 2 2  − √ 2 π z Γ  − ν 2  Φ  1 − ν 2 , 3 2 ; z 2 2  # . (8) I I I . R E S U L T S In this section we gi ve for mulas fo r th e ra w/centr al (absolu te) mom ents of a n ormal R V . If not noted otherwise, these results hold for ν > − 1 . • Raw moments: E  X ν  = ( j σ ) ν exp  − µ 2 4 σ 2  D ν  − j µ σ  (9) = ( j σ ) ν 2 ν / 2 " √ π Γ  1 − ν 2  Φ  − ν 2 , 1 2 ; − µ 2 2 σ 2  + j µ σ √ 2 π Γ  − ν 2  Φ  1 − ν 2 , 3 2 ; − µ 2 2 σ 2  # (10) = ( j σ ) ν 2 ν / 2 ·    Ψ  − ν 2 , 1 2 ; − µ 2 2 σ 2  , µ ≤ 0 Ψ ∗  − ν 2 , 1 2 ; − µ 2 2 σ 2  , µ > 0 (11) =    σ ν 2 ν / 2 Γ ( ν +1 2 ) √ π Φ  − ν 2 , 1 2 ; − µ 2 2 σ 2  , ν ∈ N 0 ev en µσ ν − 1 2 ( ν +1) / 2 Γ ( ν 2 +1 ) √ π Φ  1 − ν 2 , 3 2 ; − µ 2 2 σ 2  , ν ∈ N 0 odd . (12) • Central mo ments: E  ( X − µ ) ν  = ( j σ ) ν 2 ν / 2 √ π Γ  1 − ν 2  (13) = ( j σ ) ν 2 ν / 2 cos( π ν / 2) Γ  ν +1 2  √ π (14) =  1 + ( − 1 ) ν  σ ν 2 ν / 2 − 1 Γ  ν +1 2  √ π (15) =    σ ν ( ν − 1)!! , ν ∈ N 0 ev en 0 , ν ∈ N 0 odd . (16) • Raw absolute moments: E  | X | ν  = σ ν 2 ν / 2 Γ  ν +1 2  √ π Φ  − ν 2 , 1 2 ; − µ 2 2 σ 2  . (17) 3 • Central a bsolute moments: E  | X − µ | ν  = σ ν 2 ν / 2 Γ  ν +1 2  √ π . (18) I V . D E R I V AT I O N S In this sectio n we give d eriv ations for th e pre viou sly pr esented results. Below we use the following two identities which hold for γ ∈ R , ν > − 1 (cf. [2, Sec. 3.462]): Z ∞ −∞ ( − j x ) ν e − x 2 + j xγ dx = √ 2 − ν π e − γ 2 / 8 D ν  γ √ 2  , (19) Z ∞ 0 x ν e − x 2 − xγ dx = 2 − ( ν +1) / 2 Γ( ν + 1) e γ 2 / 8 D − ν − 1  γ √ 2  . (20) • Raw moments (9): E  X ν  = 1 √ 2 π σ 2 Z ∞ −∞ x ν exp  − 1 2 σ 2 ( x − µ ) 2  dx (21) = r 2 ν σ 2 ν π exp  − µ 2 2 σ 2  Z ∞ −∞ x ν exp  − x 2 + x µ σ √ 2  dx (22) ( 19 ) = ( j σ ) ν exp  − µ 2 4 σ 2  D ν  − j µ σ  . (23) • Central mo ments (13): Follo ws directly from ( 9) with Φ( α, γ ; 0) = 1 and, hence, D ν (0) = 2 ν / 2 √ π Γ  1 − ν 2  . (24) T o obtain (14) from (13) we use th e identity [2, S ec. 8.3 34] Γ  1 + ν 2  Γ  1 − ν 2  = π cos( π ν / 2) . (25) Then (15) follows from (14) by noting that cos( π ν / 2) = 1 + ex p( j π ν ) 2 ex p( j π ν / 2) = 1 + ( − 1 ) ν 2 j ν . (26) • Raw absolute moments (17): E  | X | ν  = 1 √ 2 π σ 2 Z ∞ −∞ | x | ν exp  − 1 2 σ 2 ( x − µ ) 2  dx (27) = r 2 ν σ 2 ν π exp  − µ 2 2 σ 2   Z ∞ 0 x ν exp  − x 2 − x µ σ √ 2  dx + Z ∞ 0 x ν exp  − x 2 + x µ σ √ 2  dx  (28) ( 20 ) = r 2 ν σ 2 ν π exp  − µ 2 4 σ 2  2 − ( ν +1) / 2 Γ( ν + 1)  D − ν − 1 ( µ/σ ) + D − ν − 1 ( − µ/σ )  (29) = r σ 2 ν 2 ν exp  − µ 2 2 σ 2  Γ( ν + 1) Γ( ν / 2 + 1) Φ  ν + 1 2 , 1 2 ; µ 2 2 σ 2  (30) = r 2 ν σ 2 ν π exp  − µ 2 2 σ 2  Γ  ν + 1 2  Φ  ν + 1 2 , 1 2 ; µ 2 2 σ 2  (31) = σ ν 2 ν / 2 Γ  ν +1 2  √ π Φ  − ν 2 , 1 2 ; − µ 2 2 σ 2  , (32) 4 where we hav e used Kummer’ s transform ation [2, Sec. 9.21 2], i.e ., Φ( α, γ ; z ) = e z Φ( γ − α, γ ; − z ) , (33) in the last s tep. • Central a bsolute moments (18): F ollows directly f rom (17) with Φ( α, γ ; 0) = 1 . R E F E R E N C E S [1] W . 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