GATE 2014 EE03 question number 27 The mean thickness and variance of silicon steel laminations are 0.2 mm and 0.02 respectively. The varnish insulation is applied on both the sides of the laminations. The mean thickness of one side insulation and its variance are 0.1 mm and 0.01 respectively. If the transformer core is made using 100 such varnish coated laminations, the mean thickness and variance of the core respectively are
 A: If $X_i$ is the thickness of the lamination $i, \ i=1 \cdots 100$ each one with a normal distribution with mean $\mu_x=0.2$ and variance $\sigma^2_x=0.02$. 
For each lamination is two  varnish insulations therefore there is $200$ insulations, denote by $Y_j$ the thickness of the insulation $j, \ j=1 \cdots 200$  each one with a normal distribution with mean $\mu_y=0.1$ and variance $\sigma^2_y=0.01$. 
Hence the thickness of the transformer is:
$$T=\sum_{i=1}^{100} X_i + \sum_{j=1}^{200} Y_j $$
Assuming the variables $X_i,Y_j$ are independent each other: 
$$E(T)=\sum_{i=1}^{100} E(X_i) + \sum_{j=1}^{200} E(Y_j)=100\mu_x + 200 \mu_y $$
$$Var(T)=Var\left(\sum_{i=1}^{100} X_i + \sum_{j=1}^{200} Y_j \right)=\sum_{i=1}^{100} Var(X_i) + \sum_{j=1}^{200} Var(Y_j)=100\sigma^2_x+200 \sigma_y^2$$
A: this my solution (and i am not 100% sure) there is a mistake in the question . It should be standard deviation in the place of variance.This problem is similar to the question in the book "Advance engg mathematics" by erwin krezig in the sec 24.3 ,problem 10 (or 9 depending on the edition).
according to a theorem in two dimensional random variable "the variance of random variable Z=X+Y is equal to sum  of variance of random variables X and Y if X and Y are independant"
going by this theorem and considering the metnioned values are standard deviation instead of variance.the variance of transformer = 100*(.02^2)+200*(.01^2)= 0.06 and standard deviation is   sqrt of variance = 0.24 mm. This corresponds to the option D in question ,which is also  given in the kEY to this question paper.
