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Asked Nov 9, 2019
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Escalate - Show every mynute steps in DETAILED!

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Expert Answer

Step 1

Given X is binomial distribution with parameters n,p.
The moment generating funciton of binomial distribution is given below

ХЕ Вiт(п. p)
M,0) - Ele* 1-(1- р)+ pе')"
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ХЕ Вiт(п. p) M,0) - Ele* 1-(1- р)+ pе')"

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Step 2

If X is a random variable with with moment generating function MX(t) then 
E[Xn] = MXn(0)
where MXn(t) is nth derivative of MX(t).
Deriving the moment generating function three times we get the following 
We use the chain rule to derivate the given moment generating function.

My)=((1-p)+pe
(a-p) + pe) - m(a-p) + pe') (pe)
M)n(a-p)pe (pe)- mn-(a-p)+ pe )
d
7-1
77
ре
dt
d
(pe
п(n
dt
+(a-p)+pe) (pe)
ME)n-(-p) + pe) (pe+n(a-p)+ pe (pe
n-1
п(п-1)
dt
ре
ре
n-2
7-3
(pe' ) +2(pe'n(n-)(1-p)+pe')
=nn-1)-2)(a-p)+ pe)
n-
1-2
+(p)+pe) (pe) + m(n- 1)(a-p) + pe)" (pe)
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My)=((1-p)+pe (a-p) + pe) - m(a-p) + pe') (pe) M)n(a-p)pe (pe)- mn-(a-p)+ pe ) d 7-1 77 ре dt d (pe п(n dt +(a-p)+pe) (pe) ME)n-(-p) + pe) (pe+n(a-p)+ pe (pe n-1 п(п-1) dt ре ре n-2 7-3 (pe' ) +2(pe'n(n-)(1-p)+pe') =nn-1)-2)(a-p)+ pe) n- 1-2 +(p)+pe) (pe) + m(n- 1)(a-p) + pe)" (pe)

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Step 3

Derivating the moment generating function for fou...

d
M )(n-n-2)((1- p)+ pe) (pe' +2(pe )}mn=1)(1-p)+pe')
n-3
n-2
dt
+n(a-p)+pe' (pe') + mn=1)((1-p)+ pe') (pe
= (n(n-1)(1-2)(1-3)(1-p) + pe') (pe)+(nn-1)(1-2)((1-p)+ pe )*° 3( p¢ )
+4(pe') n(n-(p)+ pe')+2(pe'n(n-1)(1-2)((1-p)+ pe')
7-4
1-3
n-2
+n(1-)(a-p)+pe) (pe+n(1-p)+ pe ) (pe' )
+Mn-)01-2)(a-p)+pe) (pe+mn-1)(1-p) + pe) 2(pe)
n-1
n-3
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d M )(n-n-2)((1- p)+ pe) (pe' +2(pe )}mn=1)(1-p)+pe') n-3 n-2 dt +n(a-p)+pe' (pe') + mn=1)((1-p)+ pe') (pe = (n(n-1)(1-2)(1-3)(1-p) + pe') (pe)+(nn-1)(1-2)((1-p)+ pe )*° 3( p¢ ) +4(pe') n(n-(p)+ pe')+2(pe'n(n-1)(1-2)((1-p)+ pe') 7-4 1-3 n-2 +n(1-)(a-p)+pe) (pe+n(1-p)+ pe ) (pe' ) +Mn-)01-2)(a-p)+pe) (pe+mn-1)(1-p) + pe) 2(pe) n-1 n-3

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