Beta distribution

 http://en.wikipedia.org/wiki/Beta_distribution

 

The general from of the function is:

Between a AND b with coefficient c and d

y = ((x-a)^(c-1) (b-x)^(d-1))/(b-a)^(c+d-1)


y = \frac{((x-a)^{(c-1)} (b-x)^{(d-1)})}{(b-a)^{(c+d-1)}}

y'=(x-a)^(c-2) (b-x)^(d-2) (b-a)^(-c-d+1) (a (d-1)+b (c-1)-x (c+d-2))


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y'=(x-a)^(c-1) (b-x)^(d-1) (b-a)^(-c-d+1) (a (d-1)+b (c-1)-x (c+d-2))/((x-a)(b-x))




 


MSP15911b5i1968b38ad9eb00003hb7cfgad9669c84


 


therefore the differential equation is: y’=y (a (d-1)+b (c-1)-x (c+d-2))/((x-a)(b-x))


 


 


 


MSP40742020995bbiihc29d00003i4d0b6da4g3edcf


 


 


 


 


if a=0 and b=1 it is a solution to


is a solution to this differential equation


 


 


{y}'=y\frac{(\alpha +\beta -2 )-(\alpha -1)}{(x-1)x}


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To make sure that the area below y is one and convert it to a probability density function we find the


integral_a^b (x-a)^(c-1) (b-x)^(d-1) (b-a)^(-c-d+1) dx


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which we cal B(c,d) then 


MSP12282014h5g5910iha86000014f2eeddf0b14fie


 f(x)=(1/((gamma(c)*gamma(d))/Gamma(c+d)))*((x-a)^(c-1) (b-x)^(d-1))/(b-a)^(c+d-1)





MSP24451aefi3c76eei618100003701i3hec94dd3bd 




is a probability density function with an area of 1. 

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When \alpha_1 and  \alpha_2 are not equal the curve doesn’t approach 0 asymptotically at that end but has a peak closer to that end.


This means if \alpha_1 is smaller then the peak will be close to a.


 


Beta curve with \alpha_1=1 is \alpha_2=3  b=1 f=1/B*(x-a)^0(b-x)^2


f=1/B*(x-a)^0(5-x)^2


f=(5-x)^2


is a derivative of:


f'(x)=2(x-5)


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The following is similar to y=e^-x


Exponential curve which will  be 0 at infinity and is a solution to:




When power of positive x is less than 1 and less than the power negative x the curve is monotonous decreasing.




When power of positive x is less than the power negative x the curve is skewed to right


If 1 < α < β then mode ≤ median ≤ mean. Expressing the mode (only for α, β > 1), and the mean in terms of α and β:




 \frac{ \alpha - 1 }{ \alpha + \beta - 2 } \le \text{median}  \le \frac{ \alpha }{ \alpha + \beta } ,


\text{median} \approx \frac{\alpha - \tfrac{1}{3}}{\alpha + \beta - \tfrac{2}{3}} \text{ for } \alpha, \beta \ge 1.






 


 


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In general with \alpha_1=2 is \alpha_2=3


y=((x-a)(b-x)^2)


is a solution to




with \alpha_1=2 is \alpha_2=3 a=0 b=5


y=(x(5-x)^2)


is a solution to:




In general with \alpha_1=2 is \alpha_2=3 and a=0


y=((x)(b-x)^2)


is a solution to:






In general with \alpha_1=3 is \alpha_2=2 and a=0


y=((x)^2(b-x))


is a solution to:






In general with \alpha_1=3 is \alpha_2=3


y=((x-a)^2(b-x)^2)


is a solution to




This will have a peak at (a+b)/2




In general with \alpha_1=5 is \alpha_2=5


y=((x-a)^4(b-x)^4)


is a solution to




This will have a peak at (a+b)/2


the mean is Mean=a+\frac{\alpha_1}{\alpha_1+\alpha_2}(b-a)


mode=\frac{\alpha_1-1}{\alpha_1+\alpha_2-2}(b-a))


This dynamic will be dictated by the distance from (a+b)/2, the curve will have a peak at (a+b)/2 and y will asymptotically reach 0 at x=a and x=b.


Beta distribution with \alpha_1=\alpha_2=3 is bell shaped and \alpha_1=\alpha_2=5 is very similar to normal it always has 0 skewness and kurtosis will approach 0 when \alpha_1=\alpha_2=infinity 🙂




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Below is the evolution of Beta as shape parameters change


 
















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