poisson distribution

The number of batteries sold per week by a garage may be assumed to have a poisson distribution with mean 5.

a) Find the probability that

i)exactly 6 are sold in a randomly chosen week
ii) exactly 6 are sold in each of 3 randomly chosen weeks
iii) exactly 18 are sold in a randomly chosen 3-week period.

b) Find, approximately, the probability that more than 240 are sold in a randomly chosen 52-week period.

OK firstly i write

X~Po(5)

for a-i i get this

$\displaystyle P(X=6) = \frac{e^-5 * 5^6}{6!} = 0.1462 $

is this agreed?

for part a-ii i'm not sure whether to simply do $\displaystyle (0.1462)^3$

or rewrite X~Po(15) ...?

part a-iii presents the same problem.

for part b i try

X~Po(5*52) = X~Po(260)

$\displaystyle P(x>240) = 1 - P(x<240)$

$\displaystyle Z = \frac{X-\mu}{\sigma} = \frac{240-260}{\surd{260}} = -0.39$

$\displaystyle \phi(-0.39) = 0.34827$

but i'm not sure :/

Quote:

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Originally Posted by djmccabie

The number of batteries sold per week by a garage may be assumed to have a poisson distribution with mean 5.

a) Find the probability that

i)exactly 6 are sold in a randomly chosen week
ii) exactly 6 are sold in each of 3 randomly chosen weeks
iii) exactly 18 are sold in a randomly chosen 3-week period.

b) Find, approximately, the probability that more than 240 are sold in a randomly chosen 52-week period.

OK firstly i write

X~Po(5)

for a-i i get this

$\displaystyle P(X=6) = \frac{e^-5 * 5^6}{6!} = 0.1462 $

is this agreed?

for part a-ii i'm not sure whether to simply do $\displaystyle (0.1462)^3$ Right

or rewrite X~Po(15) ...? No, then you would be ignoring the "in each of 3 weeks" part of the problem and just looking at the total number.

part a-iii presents the same problem.

for part b i try

X~Po(5*52) = X~Po(260)

$\displaystyle P(x>240) = 1 - P(x<240)$

$\displaystyle Z = \frac{X-\mu}{\sigma} = \frac{240-260}{\surd{260}} = -0.39$

$\displaystyle \phi(-0.39) = 0.34827$

but i'm not sure :/ The approach is right but your arithmetic is wrong.

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See above.

ahaa i mean $\displaystyle \phi (-1.24)$

is that correct?

for part a then would i write X~Po(15)?



appreciate the help!

Quote:

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Originally Posted by djmccabie

ahaa i mean $\displaystyle \phi (-1.24)$

is that correct?

for part a then would i write X~Po(15)?



appreciate the help!

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For part a-iii you can use a Poisson distribution with mean 15, yes.