Question

6. The gross weekly sales at a certain super market are a Gaussian random with mean $2200 and standard deviation $230. Assume that the sales from week to week are independent. (a) Find the probability that the gross sales over the next two weeks exceed $5000. (b) Find the probability that the gross weekly sales exceed $2000 in at least 2 of the next 3 weeks.

Answer 1

Answer & Let us assume that the sales over the first and the second week are represented by random variable x and y respectively. Thus, we know that x, y ~N (2200, 2302) and that t and y are independent. Let us define Z=X+Y te know that I-n (2* 2200, 2230²). This is because the sum of two normal distributions is also normal. Thus, P(Z >5000) = 1-P(Z <5000) = 1-Plze (5000-4400)| 325.2691) = 1-P (2 21.844626) - 1-0.9675 -0.0325] b) Let us define N as the number of week in the next 3 weeks where the gross weekly sales crossed 2000 We this know that: Scanned with CamScanner

NB(3,P) Pil-P(24 (2000- 92004230) Fl-PLZ 2-0.86954) 1-0.1923 0.8017 Thus, p(N>=2) = P(N=2)+p(N=3) Nowl, P(x=v) En Cy* pox qn-> Substituting these values in the equation, we get P(N=2) = 0.34635761 P(N = 3) -0.52692675 Plaaz) HP(N=3) -0.34635761 +0.52692675 -0.9032844 Scanned with CamScanner