Suppose that the demand for a company’s product in weeks 1, 2, and 3 are each no

Question

Suppose that the demand for a company’s product in weeks 1, 2, and 3 are each normally distributed and the mean demand during each of these three weeks is 50, 45, and 65, respectively. Suppose the standard deviation of the demand during each of these three weeks is known to be 10, 5, and 15, respectively. It turns out that if we can assume that these three demands are probabilistically independent then the total demand for the three week period is also normally distributed. And, the mean demand for the entire three week period is the sum of the individual means. Likewise, the variance of the demand for the entire three week period is the sum of the individual weekly variances. But be careful!

Now, suppose that the company currently has 180 units in stock, and it will not be receiving any further shipments from its supplier for at least 3 weeks. What is the probability that the company will run out of units?

Explanation / Answer

Let X denote the total demand of the three weeks , then E(x) = 50 + 45 + 65 = 160

and var(X) = 100 + 25 + 225 = 350 and standard deviation = 350^.5 = 18.7

Now you know that the 3 week period has mean 160 with standard deviation 18.7, which is enough to calculate the z-score.

Z = ( value - mean ) / sdev = ( 180 - 160 ) / 18.7 = 1.07

Look that up on your normal distribution table and you find .8577, which you interpret as 85.77% of the time 180 units will be sufficient. Company will run out 14.23% of the time .

TY!

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