Optimizing Magazine Stock Using Probability and the Central Limit Theorem

In summary, a newsagent is selling a certain magazine with a probability of 2^(-n) for n=1,2,3,... and wants to know the largest sensible number of copies to stock. Using the expectation value, the maximum profit can be found by buying 50*N magazines and selling 150*(50*s(n))-100*50N, where s(n) is the number of magazines actually sold, with s(n)=n if n<=N and s(n)=N for n>N.
  • #1
Gott_ist_tot
52
0

Homework Statement


A newsagent finds that the probability of selling 50n copies of a certain magazine is 2^{-n-1} for n = 1,2,3,... What is the largest sensible number of copies of the magazine that they should stock.


Homework Equations





The Attempt at a Solution



I really don't know how to approach this one. I hate not to provide anything in the attempt, but I am really lost on this one. Any suggestions are appreciated
 
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  • #2
I suspect they want you to compute the expectation value of the number of copies you can sell. Beyond that, what's the definition of "sensible"? Stocking "infinity" copies is always safe against demand, but you haven't stated any cost for stockpiling them.
 
  • #3
Oh, yes. He forgot to write the cost to sell them and it was amended. It is $1 for him to buy it and $1.50 is what he sells it for.

The "sensible" through me off also. Hopefully the price will help. I will start looking at the expectation.
 
  • #4
I must be missing/not understanding something here.

E[X] = 0.5(50n*2^(-n-1)) - 1.0(50n*2^(-n-1))
= -25n * 2^(-n-1)

I tried finding a maximum but I had no luck. The 50n is the number of magazines. 2^(-n-1) is the probability. Then the 0.5 is how much he would make from a sell and the -1 is how much he would lose from overstock.
 
  • #5
If he buys 50*N magazines the profit is 150*(50*s(n))-100*50N, where s(n) is the number he actually sells. s(n)=n if n<=N, s(n)=N for n>N. That's the expectation value you want to maximize as a function of N. Also 2^(-n-1) doesn't sum to unit probability for n=1,2,3... I think you want 2^(-n).
 

What is the Central Limit Theorem?

The Central Limit Theorem (CLT) is a statistical theory that states that when a large sample size is taken from a population, the sample means will be normally distributed regardless of the distribution of the population. In simpler terms, this means that as the sample size increases, the sample mean will approach the true population mean.

Why is the Central Limit Theorem important?

The Central Limit Theorem is important because it allows us to make inferences and draw conclusions about a population based on a relatively small sample size. This saves time and resources, as it is not always possible to collect data from an entire population. Additionally, the CLT is the basis for many statistical tests and models, making it a fundamental concept in statistics.

How is the Central Limit Theorem applied in real-world scenarios?

The Central Limit Theorem is applied in many real-world scenarios, particularly in inferential statistics. For example, it can be used to calculate confidence intervals, perform hypothesis testing, and make predictions about a population based on a sample. It is also commonly used in quality control and market research.

What are the assumptions of the Central Limit Theorem?

The Central Limit Theorem relies on three main assumptions: (1) the sample size is large enough (generally considered to be at least 30), (2) the observations in the sample are independent of each other, and (3) the population from which the sample is taken has a finite standard deviation. If these assumptions are met, the CLT can be applied.

Can the Central Limit Theorem be violated?

Yes, the Central Limit Theorem can be violated if the sample size is too small or if the sample is not truly random. Additionally, if the population from which the sample is taken is not normally distributed, the CLT may not hold. In these cases, alternative statistical methods may need to be used.

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