1. ## Tchebysheffs

The monthly demand for hard drives was studied for months and was found to be an average of 28 with st. dev 4. How many hard drives should be stocked at the beginning of each month to ensure that demand will exceed supply with a probability of less than .10?

Work: I first tried to standardize, but this is impossibe b/c we have no X.

2. make a sensible assumption, eg, that the total monthly demand is normal.

3. sorry guys, I still have no idea

using the 1-(1/k^2) fact we could find an interval of probability if we had some values it had to fall between. But we don't have those so it's impossible. I'm totally stuck.

4. Define: X = Total Monthly Demand

Assume $X \sim N(28,16)$
You can subsititute any other distribution assumption you prefer. The question or your teacher should have told you what sort of distributions they expect you to use.

Find the value x which satisfies
P(X \leq x) = 0.9

Usuing the methods you have been taught for the normal distribution. Dont forget to apply a continuity correction to the distribution.

5. SpringFan, can I take a stab in the dark at this? I haven't actually covered Tchebysheff's Theorem in class yet, so please tell me how far off I am.

If we let $k=\sqrt{10}$, then by theorem no more than $\frac{1}{10}$ of the values lie outside of the range $[\mu-\sqrt{10}\sigma,\mu+\sqrt{10}\sigma]$, correct?

If so, then with $\mu=28$ and $\sigma=4$ we would have no less than 90% of the values within the range $[28-4\sqrt{10},28+4\sqrt{10}]$, right?

So then we could round off to get no less than 90% within $[15,41]$, and use 41 as the answer? Granted, that's kinda fast and loose, and if we were to assume that the distribution was normal we could tighten the answer up significantly. But if we don't make any assumptions about the distribution is it possible to do better than this?

6. Originally Posted by sfspitfire23
The monthly demand for hard drives was studied for months and was found to be an average of 28 with st. dev 4. How many hard drives should be stocked at the beginning of each month to ensure that demand will exceed supply with a probability of less than .10?

Work: I first tried to standardize, but this is impossibe b/c we have no X.
We have the 2-sided Chebyshev inequality:

$pr \left( \left|\dfrac{x-\mu}{\sigma} \right|\ge k \right)\le \dfrac{1}{k^2}$

So if we want the probability to be $\le 0.1$ we take $k=\sqrt{10}$ then we have:

$\left|\dfrac{x-\mu}{\sigma} \right|\ge \sqrt{10}$

the probability of going out of stock is less than or equal to $0.1$.

So we need the larger root of:

$\left|\dfrac{x-\mu}{\sigma} \right| = \sqrt{10}$

which is $x=28+4\sqrt{10}\approx 40.65$ and since stock must be an integer this becomes $41$.

Alternativly you can use the 1-sided Chebyshev inequality:

$pr \left( \dfrac{x-\mu}{\sigma} \ge k \right)\le \dfrac{1}{1+k^2}$

which if you work this as above we get a stock of $40$

CB

7. Originally Posted by SpringFan25
make a sensible assumption, eg, that the total monthly demand is normal.