Cylindrical rods manufactured by a certain process havelengths that are normally

Question

Cylindrical rods manufactured by a certain process havelengths that are normally distributed with mean 10 cm and standarddeviation 0.1 cm. A rod is attached to a base that is exactly3 cm in length. Let X denote the length of the entireassembly, measured in cm. a. Find µx. b. Find x. c. What is the distribution of X? Cylindrical rods manufactured by a certain process havelengths that are normally distributed with mean 10 cm and standarddeviation 0.1 cm. A rod is attached to a base that is exactly3 cm in length. Let X denote the length of the entireassembly, measured in cm. a. Find µx. b. Find x. c. What is the distribution of X?

Explanation / Answer

a) X : length of assembly = l + 3
x = E(X) = E(l +3) = E(l) + 3 = 10 +3 = 13.(Remember E(x+y) = E(x) +E(y) )
b) x2= Var(l+3) = Var(l) + Var(3)+2.Co.Var (l,3) = Var(l) + 0 +2*0= Var(l) = 0.12
so x= 0.1
c) This follows normal distribution because summation of anormal variable and a constant will also follow a normal variable.Best way to convince your self is imaging p.d.f of normal curve.and think what will happen to the graph if you add a constant tothe variable. The answer is it will just shift by the constantunits to the right with out changing the shape. Hence it shouldfollow a normal distribution with new mean but same standarddeviation.
Hope this helps you.

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