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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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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 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.2012-10-27
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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.2012-10-27
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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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    Upvoted in return, thanks. I$f$ at least we had had some explanation2012-10-27