Binomial & Poisson Distributions

Printable View

September 15th 2008, 03:30 AM
ohwcomp

Binomial & Poisson Distributions

Hi, any help regarding these 2 questions is much appreciated!

Q1. I own 36 gease who each lay one golden egg in a week. I knows that 1 in 6 of these eggs is a bad egg and must be discarded. Let X be the number of good eggs I get in a week.

a) State distribution of X
b)(i)How many good eggs can I expect to get in a week?
(ii)What is the variance of the number of good eggs I get in a week?

c)(i)What is the probability that I will get 36 good eggs?
(ii)What is the probability that I get 6 bad eggs?

d) What is the probability that I get at least 34 good eggs in a week?

Q2. A recent traffic study in the town of Tanunda showed that, on average 300 cars pass through a junction between the hours of 9am to 10am.
a)What is the average number of cars per min?
b)Let X be the number of cars that passs during a one minute period. State the distribution of X and give the mean and this distribution. Now state the probability function of X.
c)Find the probability that no cars pass in a given minute.
d)Find the probability at least two cars pass in two minutes.
e)What is the expected number of cars passing in two minutes?
f)Find the probability that this expected number from part (c) actually pass through a two minute period.

Cheers.
September 15th 2008, 03:59 AM
mr fantastic

Quote:

Originally Posted by ohwcomp

Hi, any help regarding these 2 questions is much appreciated!

Q1. I own 36 gease who each lay one golden egg in a week. I knows that 1 in 6 of these eggs is a bad egg and must be discarded. Let X be the number of good eggs I get in a week.

a) State distribution of X
b)(i)How many good eggs can I expect to get in a week?
(ii)What is the variance of the number of good eggs I get in a week?

c)(i)What is the probability that I will get 36 good eggs?
(ii)What is the probability that I get 6 bad eggs?

d) What is the probability that I get at least 34 good eggs in a week?

Q2. A recent traffic study in the town of Tanunda showed that, on average 300 cars pass through a junction between the hours of 9am to 10am.
a)What is the average number of cars per min?
b)Let X be the number of cars that passs during a one minute period. State the distribution of X and give the mean and this distribution. Now state the probability function of X.
c)Find the probability that no cars pass in a given minute.
d)Find the probability at least two cars pass in two minutes.
e)What is the expected number of cars passing in two minutes?
f)Find the probability that this expected number from part (c) actually pass through a two minute period.

Cheers.

What have you been able to do? eg. Can you do Q1 a)? Where do you get stuck?
September 15th 2008, 04:13 AM
ohwcomp

Q1)
a)I get X - B (n=36 p=5/6)

b)(i)) E(X) = 30
(ii))var(X) = 5

By just using these values for n and p and putting them into the formula
E(X) = np = mean
var(X) = np(1-p)

then I get stuck on onwards

c)Do you just calculate the Binonmial Expansion with x = 36? n = 36? p=5/6?

Q2) Im having great difficulty just starting.

Please help.
September 15th 2008, 04:19 AM
mr fantastic

Quote:

Originally Posted by ohwcomp

Q1)
a) I believe I get X - B (n=35 p=5/6)

b)1) E(X) = 30
2)var(X) = 5

then I get stuck on c) onwards.
[snip]

Where is your problem calculating Pr( X = 36) and Pr(X = 30)? Just use the usual formula.
September 15th 2008, 04:21 AM
ohwcomp

Thanks!

How about d)

This least business is leaving me in trouble.

Q2) Can you help me here?

Thanks Mr. Fantastic!
September 15th 2008, 04:23 AM
mr fantastic

Quote:

Originally Posted by ohwcomp

[snip]Q2. A recent traffic study in the town of Tanunda showed that, on average 300 cars pass through a junction between the hours of 9am to 10am.
a)What is the average number of cars per min?
b)Let X be the number of cars that passs during a one minute period. State the distribution of X and give the mean and this distribution. Now state the probability function of X.
c)Find the probability that no cars pass in a given minute.
d)Find the probability at least two cars pass in two minutes.
e)What is the expected number of cars passing in two minutes?
f)Find the probability that this expected number from part (c) actually pass through a two minute period.

Cheers.

Quote:

Originally Posted by ohwcomp

[snip]
Q2) Im having great difficulty just starting.

Please help.

Can you do a)? Hint: There are 60 minutes in 1 hour.
b) What do you think the answer might be? Hint: Have you learned about the Poisson distribution?
What thoughts do you now have for c) - f)?
September 15th 2008, 04:27 AM
mr fantastic

Quote:

Originally Posted by ohwcomp

Thanks!

How about d)

This least business is leaving me in trouble.

Q2) Can you help me here?

Thanks Mr. Fantastic!

$\Pr(X \geq 34) = \Pr(X = 34) + \Pr(X = 35) + \Pr(X = 36)$ .
September 15th 2008, 05:12 AM
ohwcomp

Q1)d) Thanks for that, I realise it would be more work if you did P(X<34)!

Q2) a) I got the value of 5. LOL Talk about a mind block.

b) Poisson distribution for sure. But how do you calculate this random variable lambda?
is it from a)

d)X >= 2 how can you work this out, can u do the same thing like Q1 and split it up into individual probabilities?
e)Help pls
f)Help pls

Cheers.
September 15th 2008, 12:57 PM
mr fantastic

Quote:

Originally Posted by ohwcomp

[snip]
Q2. A recent traffic study in the town of Tanunda showed that, on average 300 cars pass through a junction between the hours of 9am to 10am.
a)What is the average number of cars per min?
b)Let X be the number of cars that passs during a one minute period. State the distribution of X and give the mean and this distribution. Now state the probability function of X.
c)Find the probability that no cars pass in a given minute.
d)Find the probability at least two cars pass in two minutes.
e)What is the expected number of cars passing in two minutes?
f)Find the probability that this expected number from part (c) actually pass through a two minute period.

Cheers.

Quote:

Originally Posted by ohwcomp

[snip]
Q2) a) I got the value of 5. LOL Talk about a mind block.

b) Poisson distribution for sure. But how do you calculate this random variable lambda?
is it from a) Mr F says: The parameter ${\color{red}\lambda}$ appearing in the Poisson probability distribution is the mean, that is, the average, of the random variable ....

d)X >= 2 how can you work this out, can u do the same thing like Q1 and split it up into individual probabilities? Mr F says: ${\color{red}\Pr(Y \geq 2) = 1 - \Pr(Y \leq 1)}$ where Y (not to be confused with X) is the random variable number of cars passing in a two minute period.

e)Help pls Mr F says: See my response to a) and b).
f)Help pls Mr F says: Calculate ${\color{red}\Pr(Y = \lambda)}$ where ${\color{red}\lambda}$ is your answer to e).

Cheers.

You need to extensively revise this section of work. The answers to all your questions rely on basic understanding of the relevant probability distributions.
September 15th 2008, 03:34 PM
awkward

Quote:

Originally Posted by ohwcomp

Hi, any help regarding these 2 questions is much appreciated!

Q1. I own 36 gease who each lay one golden egg in a week. I knows that 1 in 6 of these eggs is a bad egg and must be discarded. Let X be the number of good eggs I get in a week.

a) State distribution of X
b)(i)How many good eggs can I expect to get in a week?
(ii)What is the variance of the number of good eggs I get in a week?

c)(i)What is the probability that I will get 36 good eggs?
(ii)What is the probability that I get 6 bad eggs?

d) What is the probability that I get at least 34 good eggs in a week?

Q2. A recent traffic study in the town of Tanunda showed that, on average 300 cars pass through a junction between the hours of 9am to 10am.
a)What is the average number of cars per min?
b)Let X be the number of cars that passs during a one minute period. State the distribution of X and give the mean and this distribution. Now state the probability function of X.
c)Find the probability that no cars pass in a given minute.
d)Find the probability at least two cars pass in two minutes.
e)What is the expected number of cars passing in two minutes?
f)Find the probability that this expected number from part (c) actually pass through a two minute period.

Cheers.

ohwcomp,

You have correctly guessed (I think) that the author of Q2 expects you to say the distribution of the number of cars is Poisson.

[grump] However, in my role as curmudgeon, I wish to state that there is nothing in the problem statement to support this-- it's just a hopeful assumption. Mathematicians find it convenient to assume a Poisson distribution (and exponential inter-arrival times), because these distributions lend themselves to analytical solutions. Sometimes the convenience overrides real-world considerations. In my personal experience cars in traffic often arrive in batches, but that isn't the way a Poisson distribution behaves. [/grump]
September 15th 2008, 05:35 PM
ohwcomp

Thank you all for your help!
10 I got for Lambda.

and just subbing in x=1 x=0 and x=10 for Y and X.

Cheers again!! Much appreciated
September 15th 2008, 07:42 PM
mr fantastic

Quote:

Originally Posted by ohwcomp

Thank you all for your help!
10 I got for Lambda.

and just subbing in x=1 x=0 and x=10 for Y and X.

Cheers again!! Much appreciated

Make sure you're using $\lambda = 5$ in the distribution for X and $\lambda = 10$ in the distribution for Y.

Note: I corrected a small typo I made in post #9.
September 16th 2008, 01:27 AM
mr fantastic

Quote:

Originally Posted by awkward

ohwcomp,

You have correctly guessed (I think) that the author of Q2 expects you to say the distribution of the number of cars is Poisson.

[grump] However, in my role as curmudgeon, I wish to state that there is nothing in the problem statement to support this-- it's just a hopeful assumption. Mathematicians find it convenient to assume a Poisson distribution (and exponential inter-arrival times), because these distributions lend themselves to analytical solutions. Sometimes the convenience overrides real-world considerations. In my personal experience cars in traffic often arrive in batches, but that isn't the way a Poisson distribution behaves. [/grump]

I was going to say exactly this but didn't have the energy.