By Fluegge S. (ed.)

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Example text

By using the technique of cumulant generating functions, derive the expectation value, variance, skewness and kurtosis of the distribution of the total energy E = 12 m(Vx2 +Vy2 +Vz2 ) per particle, in the simple situation of an anisotropic Maxwellian gas (f → ∞, and g → ∞).

Furthermore, ¡ µ1 (ω)¡ ⇒ µ2 (ω) ∧ µ2 (ω) ⇒ µ1 (ω) , one has the property µ1 (ω) ⇔ µ2 (ω) = 1 for µ1 (ω) = µ2 (ω), which is phrased as ‘the (multi-valued) 20 1 Elements of Probability Theory truth value of the equivalence relation is one on the diagonal’, sc. of the square obtained by plotting µ2 (ω) against µ1 (ω). 2. The first two operators we have defined above are sometimes called strong conjunction, and weak disjunction, respectively. ) A weak conjunction is µ1 (ω) ∧w µ2 (ω) = min µ1 (ω), µ2 (ω) , and the corresponding strong dis(ω) ∨ µ (ω) = 1 − min 1 − µ1 (ω), 1 − junction (by the Morgan’s Law) is µ 1 s 2 ¡ ¡ µ2 (ω) = max µ1 (ω), µ2 (ω) ¡.

N} np Binomial B(n, p) − 6 n 2(1−f /2) Γ(f /2) 12 (g−2)(g−4) (Df,g ) f (g−6)(g−8) 6 f (f +1)(f −2g)+g(g+1)(g−2f ) fg (f +g+2)(f +g+3) 6 f 6 f −4 12 f 6 N−1 (Ca,n,N ) (N−2)(N−3) 1 µ 1+4p+p2 np 1 np(1−p) ¡n ¡ f 1 F1 ( 2 , 1 + g; s) − g2 ; − fg s) 1 F1 (f, f (1 − gs)−f 2 √ √ (|s| f )f /2 K f (|s| f ) (1 − 2s)−f /2 (1 − λs)−1 s 2 F1 (−n,−a,N−a−n+1;e ) 2 F1 (−n,−a,N−a−n+1;1) 1−p 1−pes exp µ(es − 1) (1 − p + pes )n Density, expectation, variance, skewness, (excess of) kurtosis and moment generating function of some probability distributions.

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Encyclopedia of physics, vol. 9. Fluid dynamics III by Fluegge S. (ed.)


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