poisson distribution

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May 10th 2009, 09:47 AM
djmccabie

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

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

is this agreed?

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

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

part a-iii presents the same problem.

for part b i try

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

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

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

$\phi(-0.39) = 0.34827$

but i'm not sure :/
May 10th 2009, 12:23 PM
awkward

Quote:

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

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

is this agreed?

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

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

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

$\phi(-0.39) = 0.34827$

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

See above.
May 10th 2009, 12:31 PM
djmccabie

ahaa i mean $\phi (-1.24)$

is that correct?

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

appreciate the help!
May 10th 2009, 01:36 PM
awkward

Quote:

Originally Posted by djmccabie

ahaa i mean $\phi (-1.24)$

is that correct?

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

appreciate the help!

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