CIE statistic2 doubts! Help please

[Tohru Kyo Sohma]

anyone who did CIE statistics 2 pls help me with qns 7 part (b) and qns 8 of chapter one:the poisson distribution.          pls help me

[Romeesa-Chan]

Please try uploading the image of the question yuh have doubt in! =]

[Most UniQue™]

anyone who did CIE statistics 2 pls help me with qns 7 part (b) and qns 8 of chapter one:the poisson distribution.          pls help me

Please give the link to the question paper or mention it's of which year so that others can be able to answer.


Thank You

[Tohru Kyo Sohma]

its not from pastpapers..its from the text book...i told u guys im doing self study for S2
here is the question;
Q.7)Assume that a car pass under a bridge at a rate of 100 per hour and that a Poisson distribution is appropriate.
(a)what is the probability that during a 3 min period no car will pass under the bridge?
(b)what time interval is such that the probability is atleast 0.25 that no car will pass under the bridge during that interval

Q.8)A radioactive source emits particles at an average rate of one per second. Assume that the number of emissions follows a Poisson distribution.
(a)calculate the probability that 0 or 1 particle will be emitted in 4 seconds.
(b)the emission rates changes such that the probability of 0 or 1 emission in 4 seconds becomes 0.8. What is the new emission rate?

[Alpha]

its not from pastpapers..its from the text book...i told u guys im doing self study for S2
here is the question;
Q.7)Assume that a car pass under a bridge at a rate of 100 per hour and that a Poisson distribution is appropriate.
(a)what is the probability that during a 3 min period no car will pass under the bridge?
(b)what time interval is such that the probability is atleast 0.25 that no car will pass under the bridge during that interval

"...at a rate of 100 per hour"
60 mins => 100 cars

(a) Find the mean, we denote it by for a Poisson Distribution. [Equivalent of E(X)]
    3 mins => 100/60*3 = 5 cars.

X is the random variable denoting the number of cars passing under the bridge during a 3 minutes interval.
Therefore, X ~ Po (5).

No cars -> X=0.

P (X=0) = e^(-5)*5^0/ 0!
            = e^(-5)

Give your answer to 3 significant figures.

(b) Let the required time be n minutes.

Now, X  is the random variable denoting the number of cars passing under the bridge during an n minutes interval.

= 100n/60 = 10n/6.

"...probability is atleast 0.25 that no car will pass under the bridge during that interval"

I will use 'G' to denote the greater or equal to sign: ,

and 'L' to denote the less or equal to sign: .

X ~ Po (10n/6).

P(x=0) G 0.25
e^(-10n/6) G 0.25
Take ln on both sides.
-10n/6 G ln 0.25
n L ln 0.25* (-6/10) [G changes to L because you are multiplying by a negative number]

n L 0.831

"At least", prob. should keep on increasing. Therefore, the value of e^(-10n/6) should be greater and greater, which will happen the smaller -10n/6 gets. e^(-no.)= 1/e^(no.)

Take the greatest value of n you can from the answer you obtained, i.e. n= 0.831.

Check by substituting n= 0.831 in e^(-10n/6). You will get a probability which is greater than 0.25, answer is correct.
It's advisable to check your answers for such types of questions in probability.

Question 8 is similar to question 7. Since you are self-studying, I would suggest you try question 8 by yourself by first understanding my procedures in question 7. Post your answer here so that I can check it for you. If you are still unable to do it, let me know, I'll do it then.
Hint for question 8: probability that 0 or 1 => Add P(x=0) and P(X=1).

I hope it's clear. If not, lemme know.

[Tohru Kyo Sohma]

"...at a rate of 100 per hour"
60 mins => 100 cars

(a) Find the mean, we denote it by for a Poisson Distribution. [Equivalent of E(X)]
    3 mins => 100/60*3 = 5 cars.

X is the random variable denoting the number of cars passing under the bridge during a 3 minutes interval.
Therefore, X ~ Po (5).

No cars -> X=0.

P (X=0) = e^(-5)*5^0/ 0!
            = e^(-5)

Give your answer to 3 significant figures.

(b) Let the required time be n minutes.

Now, X  is the random variable denoting the number of cars passing under the bridge during an n minutes interval.

= 100n/60 = 10n/6.

"...probability is atleast 0.25 that no car will pass under the bridge during that interval"

I will use 'G' to denote the greater or equal to sign: ,

and 'L' to denote the less or equal to sign: .

X ~ Po (10n/6).

P(x=0) G 0.25
e^(-10n/6) G 0.25
Take ln on both sides.
-10n/6 G ln 0.25
n L ln 0.25* (-6/10) [G changes to L because you are multiplying by a negative number]

n L 0.831

"At least", prob. should keep on increasing. Therefore, the value of e^(-10n/6) should be greater and greater, which will happen the smaller -10n/6 gets. e^(-no.)= 1/e^(no.)

Take the greatest value of n you can from the answer you obtained, i.e. n= 0.831.

Check by substituting n= 0.831 in e^(-10n/6). You will get a probability which is greater than 0.25, answer is correct.
It's advisable to check your answers for such types of questions in probability.

Question 8 is similar to question 7. Since you are self-studying, I would suggest you try question 8 by yourself by first understanding my procedures in question 7. Post your answer here so that I can check it for you. If you are still unable to do it, let me know, I'll do it then.
Hint for question 8: probability that 0 or 1 => Add P(x=0) and P(X=1).

I hope it's clear. If not, lemme know.

thanks alot alpha
the answer to a is correct
but the answer to b according to my textbook is 49.9s
i'll try out question 8 myself
how does the In button work?
again thanks alot
i'll +rep u soon...they're asking me to spread the love!

[Alpha]

thanks alot alpha
the answer to a is correct
but the answer to b according to my textbook is 49.9s
i'll try out question 8 myself
how does the In button work?
again thanks alot
i'll +rep u soon...they're asking me to spread the love!

X  is the random variable denoting the number of cars passing under the bridge during an n minutes interval.

n= 0.831 mins
  =0.831*60 s
  = 49.86 s
  = 49.9 s.

You're getting stressed, chill. 

You're welcome.
And it's okay. Don't worry about the +rep. Do quest.8 now. And show me what you did.

The only difficulty in S2 is that the chapters resemble each other. 

[Alpha]

And most important. 

Did you understand my explanation?

[Tohru Kyo Sohma]

And most important. 

Did you understand my explanation?

i got most of it but the part about using In button.....i dunno how to use it!

[Alpha]

i got most of it but the part about using In button.....i dunno how to use it!

Oh yes, I forgot this part.

It's ln, not In. Log e to the base of e. See on a scientific calculator. Chapter ref.: logarithms.
http://www.chem.tamu.edu/class/fyp/mathrev/mr-log.html

[Tohru Kyo Sohma]

Oh yes, I forgot this part.

It's ln, not In. Log e to the base of e. See on a scientific calculator. Chapter ref.: logarithms.
http://www.chem.tamu.edu/class/fyp/mathrev/mr-log.html

ok
i did question 8....i got part a but not part b
this is how i did it
(a)1 sec=1 particle
therefore for 4 sec =4 particles
X~Po(4)
P(X=0 or 1)=e^-4 +e^-4 *4
                =0.091
(b)P(X=0 or 1)=0.8
     e^-n +e^-n *n =0.8
       
this is all i got

[Alpha]

ok
i did question 8....i got part a but not part b
this is how i did it
(a)1 sec=1 particle
therefore for 4 sec =4 particles
X~Po(4)
P(X=0 or 1)=e^-4 +e^-4 *4
                =0.091
(b)P(X=0 or 1)=0.8
     e^-n +e^-n *n =0.8
       
this is all i got

Hm... yes, I got that too. You can use iteration for that, but I don't think it's used in Stats, I never did, at least. 
I'll think about another way to do it and let you know.

[Tohru Kyo Sohma]

Hm... yes, I got that too. You can use iteration for that, but I don't think it's used in Stats, I never did, at least. 
I'll think about another way to do it and let you know.

interation?
never heard of it
thanks alot alpha for your help!

[Alpha]

interation?
never heard of it
thanks alot alpha for your help!

Iteration. It's making n subject, and replacing with subsequent values obtained.
http://www.projectalevel.co.uk/as_a2_maths/iteration

But no, I don't think it can be used here... there must be some other way out.

You're always welcome.

[Tohru Kyo Sohma]

i have a few doubts here
Qns.1 the number of goals scored by a football team during a season gave the following results.
      _____________________________________________________
      Number of goals per match     0  1  2  3  4  5  6  7
       Number of matches              5  19 9  5  2  1  0  1
      _____________________________________________________
Calculate the mean and variance of the distribution.Calculate also the relative frequencies and the theoretical probabilities for x=0,1,2,3,4,5,6,>=7 ,assuming a Poisson distribution with the same mean.Do you think, in light of your calculations, that the Poisson distribution provides a suitable model for the number of goals per match?


Qns.2 The number of cars passing a given point in 100 10-second intervals was observed as follows.
          _________________________________________
          Number of cars       0   1   2   3   4   5
          Number of intervals  47 33 16  3   0   1
         _________________________________________
        Do you think that a Poisson distribution is a suitable model?


Qns.3 When a large number of flashlights leaving a factory is inspected it is found that the bulb is faulty in 1% of the flashlights and the switch is                   
        faulty in 1.5% of them.Assuming that the faults occur independently and at random,find
      (a)the probability that a sample of 10 flashlights contain no flashlights with a faulty bulb,
      (b)the probability that a sample of 80 flashlights contains at least one flashlight with both a defective bulb and a defective switch,
      (c)the probability that a sample of 80 flashlights contains more than two defective flashlights.


so these are my doubts
the answers to them are below
1. 1.71,1.97;
    0.12,0.45,0.21,0.12,0.05,0.02,0,0.02,0.18,0.31,0.26,0.15,0.06,0.02,0.01,0.002
    Poisson distribution is not suitable

2. mean=0.79,variance=0.87
    theoretical probabilities 0.45,0.36,0.14,0.04,0.01,0.001. Yes.

3.(a)0.904  (b)0.0119  (c)0.320

this question below,i know how to do but im a lttle confused since poisson distribution is not possible
qns.there are 5000  students in a university.Calculate the probabilty that exactly 15 of them have their birthday on 1 january, by using a suitable   Poisson distribution
here i calculate np>5 therefore poisson distribution is not possible
but the text gives answer for it 0.0965