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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