Question

A pet store sells fancy handcrafted nametags to go on dogs' and cats' collars. The store's profit is $6 for each dog nametag and $5 for each cat nametag sold. Each week the store sells an average of 12 dog tags with a standard deviation of 4 tags, and an average of 15 cat tags with a standard deviation of 3. The store's expected weekly profit on these products is $147. Assuming sales are independent, what's the standard deviation of this weekly profit?

Answer 1

We know that for two variables X and Y

Var ( aX+bY) = a2V(X) +b2V(Y) +2ab Cov (X,Y)

Cov (X, Y) = 0 if X and Y are independent

Thus , for independent variables

Var ( aX+bY) = a2V(X) +b2V(Y)

Let X be the number of dog nametag and Y be the number of cat name tag

with E(X) =12 , E(Y) =15

Profit for each dod nametag is $6 and profit for cat nametag is $5  

Total weekly Profit = 6X+5Y

Expected profit = E( 6X+5Y)=6E(X)+5(Y) = 6*12 +5*15 = 147

Expected weekly profit =$147

Var(X) = 42=16 , Var (Y) = 32 =9  

Var ( 6X+5Y) = 36 Var (X) + 25 Var (Y) = 36*16+25*9 = 801

Standard deviation of profit  =

Therefore , Standard deviation of weekly profit = $28.30