Question

You are responsible for production and sales of Bosporus takeaway snacks. Daily production of snacks is normally distributed, with a mean of 40 and a variance of 100. Daily sales are also normally distributed, with a mean of 40 and a standard deviation of 6. Sales and production have a correlation of 0.50. The selling price per snack is $7. The variable production cost per snack is $3 The fixed production costs per day are $120. What is the probability that total revenue is greater than total costs on any day?

Answer 1

$X=daily~production~of~snacks\sim N(40,10^2)\\ Y=Daily~sales\sim N(40,~6^2)\\ (X,Y)\sim Bivariate~Normal(40,40,10^2,6^2,0.50)\\ Z=total~revenue-total~cost=7Y-3X-120\sim Normal~distribution~with\\ mean=7*40-3*40-120=40\\ Variance=7^2*6^2+3^2*10^2-42*0.5*10*6=1404\\ P(Z>0)=1-\Phi((-40)/\sqrt{1404})=0.8571$