_{1}

^{*}

The statistical parameters of five generalizations of the Lindley distribution, such as the average, variance and moments, are reviewed. A new double truncated Lindley distribution with three parameters is derived. The new distributions are applied to model the initial mass function for stars.

The Lindley distribution, after [

We report some basic information on the adopted sample and on the original Lindley distribution with one parameter.

The experimental sample consists of the data x i with i varying between 1 and n; the sample mean, x ¯ , is

x ¯ = 1 n ∑ i = 1 n x i , (1)

the unbiased sample variance, s 2 , is

s 2 = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2 , (2)

and the sample rth moment about the origin, x ¯ r , is

x ¯ r = 1 n ∑ i = 1 n ( x i ) r . (3)

The Lindley probability density function (PDF) with one parameter, f ( x ) , is

f ( x ; c ) = c 2 e − c x ( x + 1 ) 1 + c ; x > 0 , c > 0 (4)

where x > 0 and c > 0 .

The cumulative distribution function (CDF), F ( x ) , is

F ( x ; c ) = 1 − ( 1 + c x 1 + c ) e − c x ; x > 0 , c > 0. (5)

At x = 0 , f ( 0 ) = c 2 1 + c and is not zero.

The average value or mean, μ , is

μ ( c ) = 2 + c c ( 1 + c ) , (6)

the variance, σ 2 , is

σ 2 ( c ) = c 2 + 4 c + 2 c 2 ( 1 + c ) 2 . (7)

The rth moment about the origin for the Lindley distribution, μ ′ r , is

μ ′ r = c − r Γ ( r + 2 ) + c 1 − r Γ ( r + 1 ) 1 + c , (8)

where

Γ ( z ) = ∫ 0 ∞ e − t t z − 1 d t , (9)

is the gamma function, see [

μ 3 = 2 c 3 + 12 c 2 + 12 c + 4 c 3 ( 1 + c ) 3 (10a)

μ 4 = 9 c 4 + 72 c 3 + 132 c 2 + 96 c + 24 c 4 ( 1 + c ) 4 (10b)

More details can be found in [

We review the statistics of the Lindley distribution with two parameters, power, generalized, new generalized and new weighted.

The Lindley PDF with two parameters TPLD [

f ( x ; b , c ) = c 2 ( b + x ) e − c x b c + 1 , (11)

where x > 0 , c > 0 and b c > − 1 . The CDF of the TPLD is

F ( x ; c , b ) = 1 − ( b c + c x + 1 ) e − c x b c + 1 . (12)

The average value or mean of the TPLD is

μ ( b , c ) = b c + 2 c ( b c + 1 ) , (13)

and the variance of the TPLD is

σ 2 ( b , c ) = b 2 c 2 + 4 b c + 2 c 2 ( b c + 1 ) 2 . (14)

The mode of the TPLD is at

M o d e = 1 − b c c , (15)

see Equation (2.3) in [

μ ′ r = c 1 − r b Γ ( r + 1 ) + c − r Γ ( r + 2 ) b c + 1 . (16)

The two parameters b and c can be obtained by the following match

μ = x ¯ (17a)

σ 2 = s 2 , (17b)

which means

b ^ = − ( s 2 + x ¯ 2 ) ( x ¯ − 2 s 2 + 2 x ¯ 2 − 2 s 2 ) ( x ¯ − 2 s 2 + 2 x ¯ 2 + x ¯ 2 − s 2 ) ( 2 x ¯ + − 2 s 2 + 2 x ¯ 2 ) , (18)

and

c ^ = 2 x ¯ + − 2 s 2 + 2 x ¯ 2 s 2 + x ¯ 2 . (19)

The power Lindley PDF with two parameters (PLD) according to [

f ( x ; b , c ) = c b 2 ( 1 + x c ) x c − 1 e − b x c b + 1 , (20)

where b, c and x > 0 . The CDF of the PLD is

F ( x ; c , b ) = ( − b x c − b − 1 ) e − b x c + b + 1 b + 1 . (21)

The average value or mean of the PLD is

μ ( b , c ) = ( b − c − 1 c + b c − 1 c c + b − c − 1 ) Γ ( c + 1 c ) ( b + 1 ) c , (22)

and the variance of the PLD is

σ 2 ( b , c ) = N A D A , (23)

where

N A = − b − 2 c − 1 ( Γ ( c + 1 c ) ) 2 c 2 + b − 2 c − 1 Γ ( c + 2 c ) b c 2 − b 2 c − 2 c ( Γ ( c + 1 c ) ) 2 c 2 − 2 ( Γ ( c + 1 c ) ) 2 b − 2 + c c c 2 + b − 2 + c c Γ ( c + 2 c ) b c 2 − 2 b − 2 c − 1 ( Γ ( c + 1 c ) ) 2 c + 2 b − 2 c − 1 Γ ( c + 2 c ) b c + b − 2 c − 1 Γ ( c + 2 c ) c 2 − 2 ( Γ ( c + 1 c ) ) 2 b − 2 + c c c + b − 2 + c c Γ ( c + 2 c ) c 2 − b − 2 c − 1 ( Γ ( c + 1 c ) ) 2 + 2 b − 2 c − 1 Γ ( c + 2 c ) c , (24)

and

D A = ( b + 1 ) 2 c 2 . (25)

The mode of the PLD is at

M o d e = − c b + 1 + ( b 2 + 4 ) c 2 + ( − 2 b − 4 ) c + 2 c − 1 2 c b . (26)

The rth moment about the origin for the PLD is

μ ′ r = b − r + c c Γ ( r + c c ) + b − r c Γ ( r + 2 c c ) b + 1 . (27)

The two parameters b and c of the PLD can be found by numerically solving the nonlinear system given by Equation (17a) and Equation (17b).

The generalized Lindley PDF with three parameters (GLD) according to [

f ( x ; a , b , c ) = b 2 ( b x ) a − 1 ( c x + a ) e − b x ( c + b ) Γ ( a + 1 ) , (28)

where a, b, c and x > 0 . The CDF of the GLD is

F ( x ; a , c , b ) = e − 1 / 2 b x ( x a / 2 ( c b a / 2 + b a / 2 + 1 ) M a / 2 , a / 2 + 1 / 2 ( b x ) + b a + 1 x a e − 1 / 2 b x ( a + 1 ) ) ( c + b ) Γ ( a + 2 ) , (29)

where M μ , ν ( z ) is the Whittaker M function, see [

μ ( a , b , c ) = a b + a c + c b ( c + b ) , (30)

and the variance of the GLD is

σ 2 ( a , b , c ) = a b 2 + 2 c b a + c 2 a + 2 c b + c 2 b 2 ( c + b ) 2 . (31)

The hazard rate function, h ( x ; a , b , c ) , of the GLD is

h ( x ; a , b , c ) = − b a + 1 x a − 1 ( c x + a ) e − b x ( a + 1 ) e − 1 / 2 b x x a / 2 ( c b a / 2 + b a / 2 + 1 ) M a / 2 , a / 2 + 1 / 2 ( b x ) + x a b a + 1 ( a + 1 ) e − b x − ( c + b ) Γ ( a + 2 ) , (32)

and

M o d e = − a b + a c + a 2 b 2 + 2 a 2 b c + a 2 c 2 − 4 a b c 2 b c . (33)

The rth moment about the origin for the GLD is

μ ′ r = Γ ( r + a ) ( b − r c a + b − r c r + b − r + 1 a ) ( c + b ) Γ ( a + 1 ) , (34)

and in particular the third moment is

μ ′ 3 = Γ ( 3 + a ) ( a b + a c + 3 c ) ( c + b ) Γ ( a + 1 ) b 3 . (35)

The three parameters a, b and c of the GLD can be obtained by numerically solving the following three non-linear equations

μ = x ¯ (36a)

σ 2 = s 2 (36b)

μ ′ 3 = x ¯ 3 . (36c)

The new generalized Lindley PDF with three parameters (NGLD) according to [

f ( x ; a , b , c ) = ( c a + 1 x a − 1 Γ ( b ) + c b x b − 1 Γ ( a ) ) e − c x ( 1 + c ) Γ ( a ) Γ ( b ) , (37)

where a, b, c and x > 0 . The CDF of the NGLD is

F ( x ; a , c , b ) = N B ( 1 + c ) Γ ( b + 2 ) Γ ( a + 2 ) , (38)

where

N B = Γ ( b + 2 ) x a c a + 1 e − c x a + Γ ( a + 2 ) x b c b e − c x b − Γ ( b + 2 ) c Γ ( a + 1, c x ) a + Γ ( b + 2 ) x a c a + 1 e − c x + Γ ( a + 2 ) x b c b e − c x − Γ ( b + 2 ) c Γ ( a + 1, c x ) + Γ ( b + 2 ) Γ ( a + 2 ) c − Γ ( a + 2 ) Γ ( b + 1, c x ) b + Γ ( b + 2 ) Γ ( a + 2 ) − Γ ( a + 2 ) Γ ( b + 1, c x ) (49)

where Γ ( a , z ) is the incomplete Gamma function, defined by

Γ ( a , z ) = ∫ z ∞ t a − 1 e − t d t , (40)

see [

μ ( a , b , c ) = a c + b c ( 1 + c ) , (41)

and the variance of the NGLD is

σ 2 ( a , b , c ) = a 2 c − 2 a b c + a c 2 + b 2 c + a c + b c + b c 2 ( 1 + c ) 2 . (42)

The rth moment about the origin for the NGLD is

μ ′ r = c − r + 1 Γ ( r + a ) Γ ( b ) + c − r Γ ( r + b ) Γ ( a ) ( 1 + c ) Γ ( a ) Γ ( b ) , (43)

and the third moment is

μ ′ 3 = Γ ( 3 + a ) Γ ( b ) c + Γ ( 3 + b ) Γ ( a ) c 3 ( 1 + c ) Γ ( a ) Γ ( b ) . (44)

The three parameters a, b and c of the NGLD are obtained by numerically solving the three non-linear Equation (36a), Equation (36b) and Equation (36a).

The new weighted Lindley PDF with two parameters (NWL) according to [

f ( x ; b , c ) = − c 2 ( 1 + b ) 2 ( 1 + x ) ( − 1 + e − c b x ) e − c x b ( c b + b + c + 2 ) , (45)

where b, c and x > 0 . The CDF of the NWL is

F ( x ; c , b ) = N C b ( c b + b + c + 2 ) , (46)

where

N C = − e − c x b 2 c x + e − c ( 1 + b ) x b c x − e − c x b 2 c − 2 e − c x b c x + e − c ( 1 + b ) x b c + e − c ( 1 + b ) x c x − e − c x b 2 − 2 e − c x b c − e − c x c x + b 2 c + e − c ( 1 + b ) x c − 2 e − c x b − e − c x c + b 2 + c b + e − c ( 1 + b ) x − e − c x + 2 b . (47)

The average value of the NWL is

μ ( a , b , c ) = b 2 c + 2 b 2 + 3 c b + 6 b + 2 c + 6 ( c b + b + c + 2 ) c ( 1 + b ) , (48)

and the variance of the NWL is

σ 2 ( a , b , c ) = N D c 2 ( b c + b + c + 2 ) 2 ( 1 + b ) 2 , (49)

where

N D = b 4 c 2 + 4 b 4 c + 4 b 3 c 2 + 2 b 4 + 18 b 3 c + 7 b 2 c 2 + 12 b 3 + 32 b 2 c + 6 b c 2 + 24 b 2 + 30 b c + 2 c 2 + 24 b + 12 c + 12. (50)

The rth moment about the origin for the NWL is

μ ′ r = N E b ( b c + b + c + 2 ) , (51)

where

N E = − ( c 1 − r b 1 − r ( 1 + b b ) − r + b − r ( 1 + b b ) − r c − r r − c − r b 2 r + c 1 − r b − r ( 1 + b b ) − r − c 1 − r b 2 + b − r ( 1 + b b ) − r c − r − c − r b 2 − 2 c − r b r − 2 c 1 − r b − 2 c − r b − c − r r − c 1 − r − c − r ) Γ ( 1 + r ) . (52)

The two parameters b and c of the NWL can be found by numerically solving the nonlinear system given by Equation (17a) and Equation (17b).

Let X be a random variable defined in [ x l , x u ] ; the double truncated (DTL) version of the Lindley PDF with one parameter, f t ( x ; c , x l , x u ) , is

f t ( x ; c , x l , x u ) = c 2 e − c x ( x + 1 ) e − c x l c x l − e − c x u c x u + e − c x l c − e − c x u c + e − c x l − e − c x u , (53)

where the effect of the double truncation increases the parameters from one to three, see [

Its CDF, F t ( x ; b , c , x l , x u ) , is

F t ( x ; b , c , x l , x u ) = N F ( ( − 1 + ( − x u − 1 ) c ) e c x l + ( 1 + ( x l + 1 ) c ) e c x u ) 2 , (54)

where

N F = − e c ( x l + x u ) ( − ( 1 + ( x l + 1 ) c ) 2 e − c ( x l − x u ) − ( 1 + ( x + 1 ) c ) ( 1 + ( x u + 1 ) c ) e c ( − x + x l ) + ( ( 1 + ( x + 1 ) c ) e c ( − x + x u ) + 1 + ( x u + 1 ) c ) ( 1 + ( x l + 1 ) c ) ) . (55)

The average value, μ t ( c , x l , x u ) , is

μ t ( c , x l , x u ) = ( 2 + ( x u 2 + x u ) c 2 + ( 2 x u + 1 ) c ) e c x l − e c x u ( 2 + ( x l 2 + x l ) c 2 + ( 2 x l + 1 ) c ) − c ( ( − 1 + ( − x u − 1 ) c ) e c x l + e c x u ( 1 + ( x l + 1 ) c ) ) . (56)

The rth moment about the origin for the DTL, μ ′ r ( c , x l , x u ) , is

μ ′ r ( c , x l , x u ) = N G ( ( 1 + ( x l + 1 ) c ) e − c x l − ( 1 + ( x u + 1 ) c ) e − c x u ) ( r + 1 ) , (57)

where

N G = − x l r / 2 e − 1 / 2 c x l ( c 1 − r / 2 + c − r / 2 ( r + 1 ) ) M r / 2 , r / 2 + 1 / 2 ( c x l ) + ( c 1 − r / 2 + c − r / 2 ( r + 1 ) ) e − 1 / 2 c x u x u r / 2 M r / 2 , r / 2 + 1 / 2 ( c x u ) + c ( r + 1 ) ( e − c x l x l r + 1 − e − c x u x u r + 1 ) . (58)

The three parameters which characterize the DTL can be found in the following way. Consider the sample of stellar masses X = x 1 , x 2 , ⋯ , x n and let x ( 1 ) ≥ x ( 2 ) ≥ ⋯ ≥ x ( n ) denote their order statistics, so that x ( 1 ) = m a x ( x 1 , x 2 , ⋯ , x n ) , x ( n ) = m i n ( x 1 , x 2 , ⋯ , x n ) . The first two parameters x l and x u are

x l = x ( n ) , x u = x ( 1 ) . (59)

The third parameter c can be found by solving the following non-linear equation

μ t ( c , x l , x u ) = x ¯ . (60)

We report the adopted statistics for four samples of stars which will be subject of fit, with the lognormal, the Lindley generalizations and the double truncated Lindley.

The merit function χ 2 is computed according to the formula

χ 2 = ∑ i = 1 n ( T i − O i ) 2 T i , (61)

where n is the number of bins, T i is the theoretical value, and O i is the experimental value represented by the frequencies. The theoretical frequency distribution is given by

T i = N Δ x i p ( x ) , (62)

where N is the number of elements of the sample, Δ x i is the magnitude of the size interval, and p ( x ) is the PDF under examination.

A reduced merit function χ r e d 2 is evaluated by

χ r e d 2 = χ 2 / N F , (63)

where N F = n − k is the number of degrees of freedom, n is the number of bins, and k is the number of parameters. The goodness of the fit can be expressed by the probability Q, see equation 15.2.12 in [

The Akaike information criterion (AIC), see [

AIC = 2 k − 2 ln ( L ) , (64)

where L is the likelihood function and k the number of free parameters in the model. We assume a Gaussian distribution for the errors and the likelihood

function can be derived from the χ 2 statistic L ∝ e x p ( − χ 2 2 ) where χ 2 has

been computed by Equation (65), see [

AIC = 2 k + χ 2 . (65)

The Kolmogorov-Smirnov test (K-S), see [

The first test is performed on NGC 2362 where the 271 stars have a range 1.47 M ⊙ ≥ M ≥ 0.11 M ⊙ , see [

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | σ = 0.5 , μ L N = − 0.55 | 37.6 | 1.86 | 0.014 | 0.073 | 0.105 |

NGC 6611 | σ = 1.03 , μ L N = − 1.26 | 71.2 | 3.73 | 1.31 × 10^{−7} | 0.093 | 0.049 |

γ Velorum | σ = 0.5 , μ L N = − 1.08 | 55.1 | 2.84 | 5.08 × 10^{−5} | 0.092 | 0.033 |

Berkeley 59 | σ = 0.49 , μ L N = − 0.92 | 54.9 | 2.82 | 5.49 × 10^{−5} | 0.11 | 6.46 × 10^{−5} |

The second test is performed on the low-mass IMF in the young cluster NGC 6611, see [

The third test is performed on γ Velorum cluster where the 237 stars have a range 1.31 M ⊙ ≥ M ≥ 0.15 M ⊙ , see [

The fourth test is performed on young cluster Berkeley 59 where the 420 stars have a range 2.24 M ⊙ ≥ M ≥ 0.15 M ⊙ , see [

Let X be a random variable defined in [ 0, ∞ ] ; the lognormal PDF, following [

PDF ( x ; m , σ ) = e − 1 2 σ 2 ( l n ( x m ) ) 2 x σ 2 π , (66)

where m is the median and σ the shape parameter. The CDF is

CDF ( x ; m , σ ) = 1 2 + 1 2 erf ( 1 2 2 ( − l n ( m ) + l n ( x ) ) σ ) , (67)

where erf ( x ) is the error function, defined as

erf ( x ) = 2 π ∫ 0 x e − t 2 d t , (68)

see [

E ( X ; m , σ ) = m e 1 2 σ 2 , (69)

the variance, V a r ( X ) , is

V a r = e σ 2 ( e σ 2 − 1 ) m 2 , (70)

the second moment about the origin, E 2 ( X ) , is

E ( X 2 ; m , σ ) = m 2 e 2 σ 2 . (71)

The statistics for the lognormal distribution for these four astronomical samples of stars are reported in

The statistics for the Lindley distribution and its generalizations are reported in the following tables:

The best fit for NGC 6611 is obtained with the Lindley PDF with one parameter, see

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | c = 2.05 | 95.57 | 5.03 | 3.36 × 10^{−12} | 0.248 | 2.93 × 10^{−15} |

NGC 6611 | c = 2.94 | 38.35 | 2.01 | 0.0053 | 0.077 | 0.161 |

γ Velorum | c = 3.18 | 90.59 | 4.66 | 5.86 × 10^{−11} | 0.322 | 3.23 × 10^{−22} |

Berkeley 59 | c = 2.76 | 149.6 | 7.76 | 6.35 × 10^{−22} | 0.323 | 5.24 × 10^{−39} |

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | b = − 0.099 , c = 4.2 | 72.94 | 3.83 | 6.8 × 10^{−8} | 0.129 | 1.76 × 10^{−4} |

NGC 6611 | b = 0.043 , c = 4.32 | 59.11 | 3.06 | 1.23 × 10^{−5} | 0.098 | 0.033 |

γ Velorum | b = − 0.035 , c = 5.81 | 67.74 | 3.54 | 5 × 10^{−7} | 0.14 | 8 × 10^{−5} |

Berkeley 59 | b = − 0.032 , c = 4.75 | 81.47 | 4.3 | 2.35 × 10^{−9} | 0.167 | 8.62 × 10^{−11} |

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | b = 2.66 , c = 2.28 | 28.87 | 1.38 | 0.128 | 0.053 | 0.39 |

NGC 6611 | b = 3.33 , c = 1.27 | 53.53 | 2.75 | 8.88 × 10^{−5} | 0.087 | 0.08 |

γ Velorum | b = 4.64 , c = 1.64 | 106.2 | 5.67 | 8.59 × 10^{−14} | 0.16 | 2 × 10^{−6} |

Berkeley 59 | b = 3.48 , c = 1.54 | 117.1 | 6.28 | 8 × 10^{−16} | 0.187 | 2.37 × 10^{−13} |

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | a = 4.80 , b = 8.38 , c = 12.01 | 37.63 | 1.86 | 0.016 | 0.064 | 0.2 |

NGC 6611 | a = 1.4 , b = 4.8 , c = 8 | 64.34 | 3.43 | 1.96 × 10^{−6} | 0.105 | 0.017 |

γ Velorum | a = 2.53 , b = 6.5 , c = 0.00046 | 83.08 | 4.53 | 1.25 × 10^{−9} | 0.15 | 2.8 × 10^{−5} |

Berkeley 59 | a = 2.2 , b = 5.09 , c = 1 | 100.6 | 5.56 | 8.6 × 10^{−13} | 0.179 | 2.93 × 10^{−12} |

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | a = 7.34 , b = 1.57 , c = 10.61 | 48.64 | 2.5 | 5.4 × 10^{−4} | 0.075 | 0.086 |

NGC 6611 | a = 3.14 , b = − 0.36 , c = 6.24 | 111.08 | 6.18 | 1 × 10^{−14} | 0.225 | 9.22 × 10^{−10} |

γ Velorum | a = 4.19 , b = 11.51 , c = 12.2 | 50 | 2.58 | 3.4 × 10^{−4} | 0.101 | 0.014 |

Berkeley 59 | a = 5.73 , b = 19.57 , c = 14.46 | 54.14 | 2.83 | 8.1 × 10^{−5} | 0.086 | 3.2 × 10^{−3} |

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | b = 0.008 , c = 3.889 | 59.72 | 3.09 | 9.85 × 10^{−6} | 0.155 | 3.33 × 10^{−6} |

NGC 6611 | b = 1.57 , c = 3.77 | 68.46 | 3.58 | 3.81 × 10^{−7} | 0.12 | 4.2 × 10^{−3} |

γ Velorum | b = 0.0027 , c = 5.86 | 79 | 4.16 | 6.2 × 10^{−9} | 0.195 | 1.86 × 10^{−8} |

Berkeley 59 | b = 0.007 , c = 5.015 | 95.13 | 5.06 | 9 × 10^{−12} | 0.19 | 4.73 × 10^{−15} |

The best fit for γ Velorum is obtained with the lognormal PDF, see

The best fit for the young cluster Berkeley 59 is obtained with the NGLD, see

The statistics for the DTL with three parameters are reported in

In this paper we explored five generalizations of the Lindley distribution as well

Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S |
---|---|---|---|---|---|---|

NGC 2362 | c = 1.61 , x l = 0.12 , x u = 1.61 | 156.7 | 8.86 | 1.75 × 10^{−23} | 0.115 | 1.2 × 10^{−3} |

NGC 6611 | c = 2.71 , x l = 0.019 , x u = 1.46 | 45.38 | 2.31 | 0.0015 | 0.061 | 0.395 |

γ Velorum | c = 4.81 , x l = 0.158 , x u = 1.317 | 45.89 | 2.34 | 1.3 × 10^{−3} | 0.064 | 0.269 |

Berkeley 59 | c = 3.93 , x l = 0.16 , x u = 2.24 | 78.57 | 4.26 | 7.73 × 10^{−9} | 0.134 | 4.35 × 10^{−7} |

Cluster | Best fit | D | P K S |
---|---|---|---|

NGC 2362 | PLD | 0.053 | 0.39 |

NGC 6611 | DTL | 0.061 | 0.395 |

γ Velorum | DTL | 0.064 | 0.269 |

Berkeley 59 | NWL | 0.086 | 3.2 × 10^{−3} |

the double truncated Lindley distribution against the lognormal distribution. For each IMF of the four clusters here analysed, the distribution which realizes the best fit is reported in

The author declares no conflicts of interest regarding the publication of this paper.

Zaninetti, L. (2020) New Probability Distributions in Astrophysics: II. The Generalized and Double Truncated Lindley. International Journal of Astronomy and Astrophysics, 10, 39-55. https://doi.org/10.4236/ijaa.2020.101004