Poisson conditions Watch

hhattiecc
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"A certain Sixth Former is late for school once a week, on average. In a half term of seven five-day weeks, lateness on more than ten occasions results in loss of privileges the following half term"

I did this question using Poisson X~Po(7), but in the mark scheme they've used a binomial distribution X~B(35,0.2).
I thought Poisson because it's talking about an average rate of lateness in a given time period, but is it Binomial because there's only 5 opportunities to be late, i.e. Poisson breaks down because the total probability isn't 1.

So does this mean that Poisson is only used when the variable is countably infinite?
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Kevin De Bruyne
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(Original post by hhattiecc)
"A certain Sixth Former is late for school once a week, on average. In a half term of seven five-day weeks, lateness on more than ten occasions results in loss of privileges the following half term"

I did this question using Poisson X~Po(7), but in the mark scheme they've used a binomial distribution X~B(35,0.2).
I thought Poisson because it's talking about an average rate of lateness in a given time period, but is it Binomial because there's only 5 opportunities to be late, i.e. Poisson breaks down because the total probability isn't 1.

So does this mean that Poisson is only used when the variable is countably infinite?
Yes, it looks like you're right. They've been sneaky because they've given what sounds like a rate but also one of the conditions for using a Binomial distribution, which you've spotted, and your last statement sounds plausible.
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rayquaza17
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I would argue you could think of the question in two ways:
35 binomial trials with the probability of being late on a certain day is 0.2 i.e X~Bin(35,0.2)
If Y is the number of times they are late in one week, then Y~Po(1) thus in 7 weeks, X~Po(7).

I would say that they are both equally correct. In fact, if we used a poisson approximation to the binomial distribution, we would get X~Po(7) in both cases!

I'm not sure what you mean by the total probability isn't 1 for poisson?
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hhattiecc
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(Original post by rayquaza17)
I would argue you could think of the question in two ways:
35 binomial trials with the probability of being late on a certain day is 0.2 i.e X~Bin(35,0.2)
If Y is the number of times they are late in one week, then Y~Po(1) thus in 7 weeks, X~Po(7).

I would say that they are both equally correct. In fact, if we used a poisson approximation to the binomial distribution, we would get X~Po(7) in both cases!

I'm not sure what you mean by the total probability isn't 1 for poisson?
I only mean that the total probability isn't 1 using a Poisson distribution to model this case (and this case only). So if you summed up the probabilities of each possibility (i.e. the number of times which they could be late in a week), they wouldn't equal 1 because in theory, with Poisson, your variable could be infinite.
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hhattiecc
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(Original post by SeanFM)
Yes, it looks like you're right. They've been sneaky because they've given what sounds like a rate but also one of the conditions for using a Binomial distribution, which you've spotted, and your last statement sounds plausible.
Ok thank you! Yeah I know, you have to watch out for things like this because through the repetitive exam style questions they ask you get sort of trained to use Poisson as soon as you read about a time period.
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