How can i find the moment generating function in the given question(?very important for an upcoming test)

the weight distribution (in grams) of green apple is normal with a mean of 100 and standard deviation of 10
the weight distribution (in grams) of red apple is normal with a mean of 150 and standard deviation of 20.

a girl walks to the market to buy 10 apples - 5 green and 5 red. she chooses randomly the apples: first the green apples and then the red ones, and puts them all in one basket(apples are in groups by the color, red with red, green with green). assuming that there's no dependability between the weight of different apples and that the weight of the empty bag is negligible. we mark in X the total weight in kilos of the full bag.

a)what is the moment creating function of x?
b)1 kg of apples(from each kind) costs 6 dollars. what is the moment creating function of the full beg?

please help me with that, it is very important for me for an upcoming test. would really appreciate if you could elaborate and explain.

thank you so much for your help!!!

1 Answer
MattyMatty
Jun 18, 2018

#"See explanation"#

Explanation:

#M_G(t) = e^(100 t + 50 t^2)#
#M_R(t) = e^(150 t + 200 t^2)#

#"a)"#
#=> M_X(t) = (M_G(t) * M_R(t))^5#
#= e^(1250 t + 1250 t^2)#
#= e^(1250 t(1+t))#

#"b)"#
#"1250 grams = 1.25 kg = 1.25*6 = 7.5 dollar"#
#"Standard deviation 50 => standard deviation 0.3 dollar"#
#=> M_{PriceBag}(t) = e^(7.5 t+0.045 t^2)#

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