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In a fabric company, the fixed probability of a machine producing a bad fabric is $p$ independent on the fabric previously produced. At the output of the machine, $n$ fabrics are taken at random. $X$ is the random variable equal to the number of bad fabrics in a selection of $n$ fabrics at the mouth of the machine.

  • What probability distribution law does $X$ follow and why?
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3 Answers

up vote 2 down vote accepted

This fits perfectly the definition of a binomial distribution. There are $n$ independent trials, and in each trial the probability of "success," if you can call it success, is $p$. It is exactly like tossing a funny coin $n$ times, with the probability of a head equal to $p$, and our random variable the number of heads.

To put it another way, let $X_i=1$ if the $i$-th fabric is defective, and $0$ if it is not. Then $$X=\sum_{i=1}^n X_i,$$ and the sum of $n$ independent identically distributed Bernoulli random variables has binomial distribution.

Remark: There is some connection with the Poisson. If $n$ is large and $p$ is small, with $np$ of modest size, then the binomial distribution of this problem is well-approximated by the Poisson with parameter $\lambda=np$. But it is still a binomial.

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I understand this, but in my question, one only needs the number of defective fabrics whatever the order of arrangement. Unless what was considered was the order at which the fabrics come out of the machine? –  user31280 Oct 27 '12 at 1:29
    
The Binomial distribution measures the number of successes. The probability that $X=k$ is $\binom{n}{k}p^k(1-p)^{n-k}$. It is true that when we derive this formula, we consider all the possible orders in which the successes and/or failures could occur. –  André Nicolas Oct 27 '12 at 1:34
    
so where does this "order" that was considered falls in my question. Is it at the selection of the fabrics from the mouth of the machine or is it the order at which the fabrics came out of the machine? –  user31280 Oct 27 '12 at 1:36
    
It is an arbitrary order. After we have chosen our $n$ fabrics, we line them up in a row without testing them for quality. But remember, this is how we derive the formula. Once we have derived the formula, we need not consider order at all. –  André Nicolas Oct 27 '12 at 1:43
    
I guess I just have to accept it that way. Even if the order is arbitrary, it is at least acceptable. Thank you for your answer and time. –  user31280 Oct 27 '12 at 1:46
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Binomial would be more appropriate here. See the machine has a Bernoulli trial that has succes probability $p$. Because of independance, each of the $n$ fabrics you took have a $p$ chance of being bad. The number of bad fabrics is then given by the binomial distribution.

Poisson would give you an approximation of this, but $n$ would have to be large enough, and $np$ small enough.

Poisson distribution are more related to problems of the type : A machine produces fabrics at random. In $5$ minutes, it produces on average $3$ fabrics. What is the probability that you have $5$ bad fabrics after $10$ minutes?

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but doesn't binomial always consider the arrangement of the fabric? –  user31280 Oct 27 '12 at 1:22
    
What do you mean by arrangement? –  Jean-Sébastien Oct 27 '12 at 1:24
    
sorry. what i meant was that the order of selection is put into consideration when using binomial. –  user31280 Oct 27 '12 at 1:25
    
Unless you know any kind of information about the fabrics before taking your sample, you can imagine your sample as soon as it come out of the machine, as i the experience had just been realized. If however you wait the end of the day, you know that the machine produced say $k$ bad fabrics and you take a sample of $n$,then $X$ will be Hypergeometric –  Jean-Sébastien Oct 27 '12 at 1:28
    
I understand you but you didn't answer my question. When binomial is used, the order of selection is always considered which I don't see in this case. –  user31280 Oct 27 '12 at 1:33
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Defective vs. not defective usually implies binomial. If $n$ is sufficiently large and $p$ (probability of success) is very small, the Poisson distribution can be used to approximate the binomial distribution.

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Use binomial when each observation is one of two possible outcomes, e.g. a random sample of size $n$ is drawn and the number of defects found in the sample is recorded. Use Poisson when dealing with rates of occurrence e.g. if the number of defects per length of fabric exceeds $L$ then etc. –  glebovg Oct 27 '12 at 1:30
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