A machine manufactures cubes with a side length which varies Uniformly over the interval $[0.2, 0.3]$ in millimeters.
For the following problems, make sure you use the correct units. (Assume the
sides of the base and the height are all the same.)
a. What is the expected side length?
b. What is the standard deviation of the side length?
c. What is the expected area of one of the square bases?
d. What is the standard deviation of one of the square bases?
e. What is the expected volume of one of the cubes?
f. What is the standard deviation of the volume of one of the cubes?
g. What is the expected cost for making 1 cube?
h. What is the variance in the cost for making 1 cube?
i. What is the expected cost for making 10 cubes?
j. What is the variance in the cost for making 10 cubes?
a and b are simple
$E(X) = \cfrac{0.2+0.3}{2} = .25$ mm
$Var(X) = \cfrac{(0.3-0.2)^2}{12} = 1/1200$ so $sd(X) = \sqrt{1/1200} = .0289$ mm
I think I understand how to do c)
$E(area) = .25^2 = .0625 mm^2$ since we assume the sides are all the same so just square E(X)
I don't know how to do d)
I think I know how to do e)
$E(volume) = .25^3 = 1/64 mm^3$ since we assume the sides are all the same so just cube E(X)
I don't know how to do f)
g) $E(.12\cdot Volume + .06) = .12E(Volume) + .06 = .12 \cdot 1/64 + .06 = .061875$ cents
h) I don't know variance of volume so I can't do h)
i) $10 \cdot E(.12\cdot Volume + .06) = 10 \cdot (.12E(Volume) + .06) = 10 \cdot (.12 \cdot 1/64 + .06) = .61875$ cents
j) I don't know variance of volume so I can't do j)
So what I'm asking is how to find the variance for the area and volume of these cubes? Any hints would be appreciated thank you! Also I would appreciate it if someone could tell me if my methodology and answers for everything else is correct!
Thank You!
Best Answer
a. ok
b. ok
c. Wrong.
$$\mathbb{E}[X^2]=\mathbb{V}[X]+\mathbb{E}^2[X]\approx 0.0633 \text{ mm}^2$$
d. First derive the distribution of the area
$$Y=X^2$$
$$f_Y(y)=\frac{5}{\sqrt{y}}\cdot \mathbb{1}_{[0.04;0.09]}(y)$$
The area is expressed in $\text{mm}^2$ so also its st dev is. (I think you can calculate any indicator you want given the density: mean, variance, st dev etc etc)
e. Wrong. Same solution as c. You have to calculate $\mathbb{E}[X^3]$
f. Same procedure as d. First derive the distribution of $Z=X^3$ then calculate whatever you want. Alternatively you can calculate $V[X^3]$ with the distribution of $X$ as in the previous e. The best choice is up to you.
g. and following: in the text you posted there are no information about costs...