let X has distribution function given by :
F(x)=1- e^(-4x) , x>0
find :
(1) p(x<=1)
(2) p(x=4)
(3) the mean of X
(4) p (x>=25)
(5) the second moment about the mean
my solutions :
(1) 1-(e^-4)
Mr F says: Correct.
(2) zero since the distribution is continuous
Mr F says: Correct.
(3)mu=(1\16)
Mr F says: I get 1/4. Note that the pdf is , x > 0, and zero otherwise. Note also the result you get when you differentiate the moment generating function (below) and substitute t = 0 ......
(4)p (x>=25)= 1 - p (x<=25)
= 1- { 1 - e^(-4*25) }
Mr F says: Correct. And I'm sure you can simplify it.
(5) moment generating function is :
( 1 - (t\4) )^-1
Mr F says: Correct.
we differentiate it twice and substitute t=0
Mr F says: This gives you the second moment about the origin. However, the mean is not equal to zero and so this will not equal the second moment about the mean.
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