I have the following exercise and I do not know if I have raised it well in the first 2 sections and section 3 I do not know how to start.
A freight train loads containers of three types: (a) of 36 dm3, (b) of 54 dm3 and (c) of 216 dm3. Let Xi be the number of type I containers shipped on the train. Suppose that the random variables Xi are independent with normal distribution and that mean a = 200, sigma a = 9.6; mean b = 250, sigma b = 4.8 and mean c = 100, sigma c = 1.5.
Define the random variable V = total volume shipped in the train.
1) Express the variable V as a function of the variables Xi. What is the probability model of V?
2) What is the probability that the total volume shipped on the train is at most 44000 dm3?
3) A random sample of the volumes shipped on sixteen trains is taken. What is the probability that the average volume shipped on these trains is between 42100 and 42400 dm3?
For section 1) I have:
Y = 36Xa + 54 Xb + 216 Xc
N (mean y, sigma y)
Being mean Y = E (36Xa + 54 Xb + 216 Xc) = 42300
Being sigma Y = Var (36Xa + 54 Xb + 216 Xc) = $ 36^2 Var(Xa)^2 + 54^2 Var(Xb)^2 + 216^2 Var(Xc)^2 $ = 291600
Then now obtained the values, I calculate the normal of N (42300, 291600)
For question 2) Pr (V> 44000), calculating it from the normal N (42300, 291600) of section 1 for variable 43999.
For question 3) I have to recalculate the mean and variance of 16 selected trains but I would be left to variables and not numbers?
Comments:
Your expected value of $E[V]=423000$ seems to be ten times the correct figure and the magnitudes in questions 2 and 3
For question 3, if you take $n$ independent samples from a normal distribution with expected value $\mu$ and variance $\sigma^2$ then their sum is normally distributed with expected value $n\mu$ and variance $n\sigma^2$, so their average is normally distributed with expected value $\mu$ and variance $\frac{\sigma^2}{n}$