1. ## What's the probability?

You need to decide how many shelving units to allocate to a certain product. The product comes in a box and you only can fit 10 of these boxes onto shelving unit. Shipments are following normal distribution. Mean 90 boxes/month, standard deviation 8 boxes/month. Boxes arrive during the first week of the month, and then are shipped the last three weeks of the month.

a) Whats the probability that 99 or more boxes arrive in 5 or more of the next 6 months?

b) whats the likelihood that more than 610 boxes arrive over a six month period?

2. Originally Posted by amity
You need to decide how many shelving units to allocate to a certain product. The product comes in a box and you only can fit 10 of these boxes onto shelving unit. Shipments are following normal distribution. Mean 90 boxes/month, standard deviation 8 boxes/month. Boxes arrive during the first week of the month, and then are shipped the last three weeks of the month.

a) Whats the probability that 99 or more boxes arrive in 5 or more of the next 6 months?
What is the probability $p$ that $99$ or more boxes arrive in a month? Use the normal distribution means $90$ sd $8$ to find this.

The number of times that $99$ or more arrive in $6$ month has a binomial distribution $B(p,6)$, use this to calculated the probability that $99$ or more boxes arrive in $5$ or more of the next $6$ months:

$P(5+)= b(5;p,6) + b(6;p,6)$

where $b(n;p,N)$ is the binomial probability of $n$ occurences in $N$ trials with probability of success on a single trial of $p$.

b) whats the likelihood that more than 610 boxes arrive over a six month period?
The number of arrivals in a six month period has a normal distribution with mean $6\times 99$, and SD $\sqrt{6} \times 8$.

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3. thank you.
Why did you multiply mean by 6 and standard deviation by sqrt of 6?
i know it is some form of transformation. But can you clearify on a concept?
thanks

4. Originally Posted by amity
thank you.
Why did you multiply mean by 6 and standard deviation by sqrt of 6?
i know it is some form of transformation. But can you clearify on a concept?
thanks
Because the 6 month arrivals is the sum of 6 1 month arrivals and the mean of a sum is the sum of the means, and the variance of a sum is the sum of the variances. As the means and variances are all the same the mean arrivals in 6 months is 6 times the mean arrivals in 1 month, and the variance in the number of arrivals in 6 months is 6 times the variance for 1 month. The standard deviation is the square root of the variance so the 6 moth SD is sqrt(6) times the 1 month SD.

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