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Probability Distributions and Combination of Random Variables Appendix A Probability Distributions and Combination of Random Variables A.1 Probability Density Functions Normal (Gaussian) Distribution Fig. A.1 Probability density μ σ function of normal =0, =1 0.8 μ σ (Gaussian) random variables =0, =3 for different values of the 0.7 μ=0, σ=0.5 μ σ μ=2, σ=2 parameters and 0.6 ) σ 0.5 , μ 0.4 p(x| 0.3 0.2 0.1 0 −6 −4 −2 0 2 4 6 x The stating point of most of the probability distributions that arises when dealing with noise in medical imaging is the Normal distribution, also known as Gaussian distribution. Let X be a random variable (RV) that follows a normal distribution. This is usually represented as follows: X ∼ N(μ, σ2) where μ is the mean of the distribution and σ is the standard deviation. © Springer International Publishing Switzerland 2016 275 S. Aja-Fernandez´ and G. Vegas-Sanchez-Ferrero,´ Statistical Analysis of Noise in MRI, DOI 10.1007/978-3-319-39934-8 276 Appendix A: Probability Distributions and Combination of Random Variables PDF: (Probability density function) 1 (x − μ)2 p(x|μ, σ) = √ exp − (A.1) σ 2π 2σ2 MGF: (Moment generating function) σ2t2 M (t) = exp μt + X 2 Main parameters: Mean = μ Median = μ Mode = μ Variance = σ2 Main moments: μ1 = E{x}=μ 2 2 2 μ2 = E{x }=μ + σ 3 3 2 μ3 = E{x }=μ + 3μσ 4 4 2 2 4 μ4 = E{x }=μ + 6μ σ + 3σ Rayleigh Distribution Fig. A.2 Probability density 0.7 function of Rayleigh random σ=1 variables for different values 0.6 σ=2 of the parameter σ σ=3 0.5 σ=4 ) 0.4 σ p(x| 0.3 0.2 0.1 0 0 2 4 6 8 x Appendix A: Probability Distributions and Combination of Random Variables 277 The Rayleigh distribution can be seen as the distribution that models the square root of the sum of squares of two independent and identically distributed Gaussian RV with the same σ and μ = 0. This is the case, for instance of the modulus of a complex Gaussian RV with independent components: (σ) = 2 + 2 ∼ ( , σ2) R X1 X2, Xi N 0 2 2 2 R(σ) = (X1 − μ1) + (X2 − μ2) , Xi ∼ N(μi , σ ) R(σ) =|X|, X = N(0, σ2) + jN(0, σ2) PDF: x x2 p(x|σ) = exp − (A.2) σ2 2σ2 MGF: π σ σ2t2/2 t MX (t) = 1 + σ te erf √ + 1 2 2 Raw moments: k k/2 μk = σ 2 Γ(1 + k/2) Main parameters: π Mean = σ 2 Median = σ log 4 Mode = σ 4 − π Variance = σ2 2 Main moments: π μ = σ 1 2 μ = σ2 2 2 π μ = 3 σ3 3 2 4 μ4 = 8 σ 278 Appendix A: Probability Distributions and Combination of Random Variables Rician Distribution Fig. A.3 Probability density 0.6 function of Rician random A=1, σ=1 σ variables for different values 0.5 A=4, =1 of the parameters A and σ A=1, σ=2 A=1, σ=3 0.4 A=2, σ=4 ) σ 0.3 p(x|A, 0.2 0.1 0 0 2 4 6 8 x The Rician distribution can be seen as the distribution that models the square root of the sum of squares of two independent and identically distributed Gaussian RV with the same σ. ( , σ) = 2 + 2, ∼ (μ , σ2) R A X1 X2 Xi N i 2 2 R(A, σ) =|X|, X = N(μ1, σ ) + jN(μ2, σ ) PDF: x − x2+A2 Ax p(x|A, σ) = e 2σ2 I u(x), (A.3) σ2 0 σ2 where = μ2 + μ2. A 1 2 For A = 0 the Rician distribution simplifies to a Rayleigh. Raw moments 2 k k/2 A μ = σ 2 Γ(1 + k/2) L / − k k 2 2σ2 where Ln(x) is the Laguerre polynomial. It is related to the confluent hypergeo- metric function of the first kind: Ln(x) = 1 F1(−n; 1; x). The even moments become simple polynomials. Appendix A: Probability Distributions and Combination of Random Variables 279 Main moments: π A2 μ = L / − σ 1 2 1 2 2σ2 μ = A2 + 2 σ2 2 π 2 A 3 μ = 3 L / − σ 3 2 3 2 2σ2 4 2 2 4 μ4 = A + 8 σ A + 8 σ πσ2 2 2 2 2 A Var = 2σ + A − L / − 2 1 2 2σ2 1 1 A2 ≈ σ2 1 − − + O(x−3) with x = 4x 8x2 2σ2 Series Expansion of Hypergeometric Functions √ 2 x 1 1 −5/2 L / (−x) = √ + √ √ + √ + O x 1 2 π 2 π x 16 πx3/2 √ 3/2 4x 3 x 3 1 −5/2 L / (−x) = √ + √ + √ √ + √ + O x . 3 2 3 π π 8 π x 32 πx3/2 Central Chi Distribution (c-χ) Fig. A.4 Probability density 0.7 function of c-χ random σ=1, L=1 variables for different values 0.6 σ=1, L=2 of the parameters L and σ σ=2, L=1 0.5 σ=2, L=2 0.4 ,L) σ 0.3 p(x| 0.2 0.1 0 0 2 4 6 8 x 280 Appendix A: Probability Distributions and Combination of Random Variables The central-χ (c-χ) distribution can be seen as the distribution that models the square root of the sum of squares of several independent and identically distributed Gaussian RV with the same σ and μ = 0: 2L ( , σ) = 2 ∼ ( , σ2) R L Xi Xi N 0 i=1 2L 2 2 R(L, σ) = (Xi − μi ) Xi ∼ N(μi , σ ) i=1 L 2 2 2 R(L, σ) = |Yi | Yi = N(0, σ ) + jN(0, σ ). i=1 PDF: 1−L 2L−1 2 x − x2 p(x|σ, L) = e 2σ2 (A.4) Γ(L) σ2L Note that this definition of the (nonnormalized) PDF uses parameters related to MRI configuration. In other texts, m = 2L are used instead. The Rayleigh distribution is a special case of the c-χ when L = 1. Raw moments: Γ(k/2 + L) μ = σk 2k/2 . k Γ(L) For n an integer, Γ(n) = (n − 1)! and √ 1 π n Γ n + = (2k − 1) 2 2n k=1 Main moments: √ Γ(L + 1/2) μ = 2 σ 1 Γ(L) 2 μ2 = 2 L σ Appendix A: Probability Distributions and Combination of Random Variables 281 √ Γ(L + 3/2) μ = 3 2 σ3 3 Γ(L) 4 μ4 = 4 L(L + 1) σ Γ(k/2 + L) 2 Var = 2σ2 L − Γ(L) Noncentral Chi Distribution (nc-χ) Fig. A.5 Probability density 0.6 function of nc-χ random σ=1, L=1, A=1 variables for different values σ=1, L=2, A=1 0.5 of the parameters A, L and σ σ=2, L=1, A=2 σ=2, L=2, A=2 0.4 σ=1, L=1, A=3 ,L) σ 0.3 p(x|A, 0.2 0.1 0 0 2 4 6 8 x The noncentral-χ (nc-χ) distribution can be seen as the distribution that model the square root of the sum of squares of several independent and identically distributed Gaussian RV with the same σ: 2L = 2 ∼ (μ , σ2) R Xi Xi N i i=1 L 2 2 2 R = |Yi | Yi = N(μ1,i , σ ) + jN(μ2,i , σ ) i=1 PDF: − 1 L x2+A2 AL L − L AL x p(x|A , σ, L) = x e 2σ2 I − u(x), (A.5) L σ2 L 1 σ2 282 Appendix A: Probability Distributions and Combination of Random Variables with L 2 AL = |μi | . i=1 When AL = 0 the nc-χ simplifies into a c-χ. The Rician distribution is a special case of the nc-χ for L = 1. Main moments: √ Γ(L + 1/2) 1 A2 μ = 2 F − , L, − L σ 1 Γ(L) 1 1 2 2σ2 μ = A2 + 2 Lσ2 2 L √ Γ(L + 3/2) 3 A2 μ = 3 2 F − , L, − L σ3 3 Γ(L) 1 1 2 2σ2 μ = 4 + ( + ) 2 σ2 + ( + ) σ4 4 AL 4 L 1 AL 4L L 1 Γ(L + 1/2) 2 1 Var = 2σ2 1 + x − F 2 − , L, −x Γ(L) 1 1 2 1 − 2L 3 − 8L + 4L2 A2 ≈ σ2 1 + + + O(x−3) with x = 4x 8x2 2σ2 Central Chi Square Distribution (c-χ2) Fig. A.6 Probability density 0.2 function of c-χ2 random σ=1, L=2 variables for different values σ=1, L=3 of the parameters L and σ σ=2, L=2 0.15 σ=2, L=4 ,L) σ 0.1 p(x| 0.05 0 0 10 20 30 40 50 x The central-χ square (c-χ2) distribution can be seen as the distribution that models the square root of the sum of several independent and identically distributed Gaussian RV with the same σ and μ = 0: Appendix A: Probability Distributions and Combination of Random Variables 283 2L ( , σ) = 2 ∼ ( , σ2) R L Xi Xi N 0 i=1 2L 2 2 R(L, σ) = (Xi − μi ) Xi ∼ N(μi , σ ) i=1 L 2 2 2 R(L, σ) = |Yi | Yi = N(0, σ ) + jN(0, σ ). i=1 PDF: − 1 x L 1 − x p(x|σ, L) = e 2σ2 u(x), (A.6) 2σ2Γ(L) 2σ2 Raw moments: Γ(k + L) μ = σ2k 2k .
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