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s2 poisson vs normal distribution (quick question) Watch

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    A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.


    The café serves breakfast every day between 8 a.m. and 12 noon.

    Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.


    i worked out that mean = (1/6)(4)(60) = 40 (basically, probability in 1 minute x 4 x 60 minutes - since it's for 4 hours)

    variance is therefore (1/6)(4)(60)(5/6) = 33.3... (variance is np(p-1), right?)

    so Y - N[40,33.3] is the model i got

    however in the mark scheme they only calculated the mean and then modelled it as Y - N[40,40].

    is mean ALWAYS equal to variance in a poisson distribution? if not, how can you tell when the mean IS equal to variance? would really appreciate any input, thanks in advance
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    If X \sim \text{Po}(\lambda) then the mean and variance are both \lambda.

    np(1-p) is for the binomial distribution.
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    (Original post by A-level Jim)
    i worked out that mean = (1/6)(4)(60) = 40 (basically, probability in 1 minute x 4 x 60 minutes - since it's for 4 hours)
    Correct. Only I'd have done it slightly simpler, by just saying 10x4=40, since 1 customer in 6 minutes ==> 10 customers in 60 minutes=1 hour ==> 10x4=40 customers in 4 hours.

    variance is therefore (1/6)(4)(60)(5/6) = 33.3... (variance is np(1-p), right?)
    Only for binomial distributions. Here you are using the normal distribution to approximate a poission distribution (since "n is large and p is not near 0 or 1"). And for poisson distributions, the mean and variance are equal.

    ...

    however in the mark scheme they only calculated the mean and then modelled it as Y - N[40,40]. -- Because they are equal!

    is mean ALWAYS equal to variance in a poisson distribution? if not, how can you tell when the mean IS equal to variance? would really appreciate any input, thanks in advance
    Yes. It is.
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    (Original post by A-level Jim)
    A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.


    The café serves breakfast every day between 8 a.m. and 12 noon.

    Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.


    i worked out that mean = (1/6)(4)(60) = 40 (basically, probability in 1 minute x 4 x 60 minutes - since it's for 4 hours)

    variance is therefore (1/6)(4)(60)(5/6) = 33.3... (variance is np(p-1), right?)

    so Y - N[40,33.3] is the model i got

    however in the mark scheme they only calculated the mean and then modelled it as Y - N[40,40].

    is mean ALWAYS equal to variance in a poisson distribution? if not, how can you tell when the mean IS equal to variance? would really appreciate any input, thanks in advance
    The question mentions rate which, in most of the questions I've seen, implies a poisson distribution
 
 
 
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