H2 maths tutor needs help

• Mad Hat

  19 Jul 10, 11:59

  Wen Shih, thanks, but I don't understand your reply

  Eagle, I take your point that the current question does not state that the car arrivals follow a Poisson distribution. But what happens if this is explicitly stated? Consider this modified question:

  At a certain junction, cars arrive randomly (following a Poisson distribution) at an average rate of 7 per minute, while other vehicles arrive randomly (also following a Poisson distribution) at an average rate of 3 per minute. So vehicles arrive randomly (following a Poisson distribution) at an average rate of 10 per minute and 70 percent of these vehicles are cars.

  (ii) Find the variance of the number of cars that arrive in a 3-hour period.

  This question contains the same info as the original, plus some added some details of my own. Here, the following calculation would be correct, yes?:

  Let X = number of cars arriving at the junction in 3 hours
  ⇒ X ~ Po(1260)
  ⇒ Var(X) is 1260.

  But what about the other calculation? Why does it fail this time when it worked the last time?

  Let X = number of vehicles arriving at the junction in 3 hours
  ⇒ X ~ Po(1800)
  ⇒ Var(X) = 1800
  ⇒ Var(0.7X) = 0.7 x 0.7 x 1800 = 882.

  In both cases, 70% of the vehicles arriving are cars, so why does this calculation work in the previous case only, but not in the present case?

• eagle

  19 Jul 10, 13:11

  Hi Mad Hat,

  in the original case, we do not know the distribution of the number of cars arriving.We do not know if it's poisson. Poisson has an addictive property, but I don't think it is stated that it can be splitted like what you said. Correct me if I'm wrong as I'm not 100% sure.

  i.e. X~Po(2) and Y~Po(3), it means X+Y~Po(5)
  But if X+Y~Po(5) and X is 0.4(X+Y), it does not mean X~Po(2) and Y~Po(3).

  70% of the vehicles arriving indicates that that is the expected value of the number of cars arriving. Unless the distribution of the cars arriving follow a poisson distribution as well, then we can say that the variance is equal to the mean.

   

  In summary, X~Po(1260) could already be wrong in the original question.

  But X~Po(1260) is correct in the modified question.

• Mad Hat

  19 Jul 10, 17:24

  Hi Eagle, yes I think I agree with your point that X ~ Po(1260) could already be wrong in the original question. The reason you gave is quite convincing.

  But I still don't understand the logic of the ACJC model answer. Why does the ACJC model answer work for the original question but not for the modified question?

• eagle

  19 Jul 10, 18:04

  So sorry... I suddenly realised I should have used independent events to explain.

  In your modified question, you are defining the poisson distribution for X specifically. You are specifically defining X as an independent event in itself.

  In the original question, it has been said that "Traffic counts at this location indicate that 70 percent of these vehicles are cars." The number of cars is dependent on the number of vehicles arriving. It is not an independent event, nor is it defined as one.

  Hope this is clearer ")

• Mad Hat

  19 Jul 10, 20:06

  Hi Eagle, thanks a lot. I think the original question is a bit misleading in that case. Because, in real life, at a traffic junction, how can anyone guarantee that the number of cars arriving is always exactly 70% of the total number of vehicles arriving, all the time? It's not realistic?

  But if that is really the intended meaning, then I guess I can accept that C = 0.7X, and therefore Var(C) = 0.49Var(X).

  C = number of cars arriving
  X = total number of vehicles arriving

• eagle

  20 Jul 10, 11:00

  Hi Mad Hat,

  it's not exactly 70%, which is why the variance is not zero :)
  It's just an average.

   

  In addition, we should not take questions in A levels at face value. Even in A level physics, there are many questions which do not fit theory. However, what is important is the concepts behind, and the method to tackle the question according to what the setter wants and expects.

• Mad Hat

  20 Jul 10, 13:29

  C = number of cars arriving
  X = total number of vehicles arriving

  Hi Eagle, but if it's just an average, then the only information we are given in the original question is that E(C) = 0.7E(X).

  From this, we cannot infer that C = 0.7X, can we?? Note that the ACJC model answer definitely makes this inference, because its next step is Var(C) = 0.7² Var(X).

• eagle

  20 Jul 10, 13:37

  Hi Mad Hat,

  if C = 0.7X, E(C) = E(0.7X) = 0.7 E(X)

  But if Var(C) will be 0.49Var(X)

  Saying C = 0.7X means that C is dependent on X. Hence Var(C) = 0.7² Var(X).

• Mad Hat

  20 Jul 10, 17:25

  Hi Eagle, I think I'm not managing to get my point across. Let me think of a clearer way to make my point..
• Mad Hat

  8 Sep 10, 15:03

  Can someone show me how to use the TI-84 graphic calculator to find the exact intersection points of two curves? For example, intersection of:
  
  y = 3|x + 1| and
  y = ¼x³ − x² − 6x + 15

  Should I graph both curves on one diagram and ask the calculator to find the exact intersection points? (What function button to press?)

  or:

  Should I combine the curves like this: y = 3|x + 1| − (¼x³ − x² − 6x + 15) and then ask the calculator to find the intersection points with the x-axis? (Again, what function button to press?)

  My student doesn't know how to do it and I don't know how to help her because I don't have this calculator model.

• eagle

  8 Sep 10, 16:14

  Hi,

  I don't think you can get the exact value with any GC if it is in terms of square roots or ln or etc type of weird non-rational values

  For TI-84, I believe you can press [2nd] [Calc], and number 5, finding intersection. The calc button is somewhere on the small blue text at the top of the calculator, around the window key I think.

  I don't own any GC, nor do I have one with me right now, so pardon me if I'm wrong.

• Terminator Hitman

  8 Sep 10, 16:21

  hey i remember ...the graphs to plot......maths was like a jigsaw puzzle game to me because i didnt have computer or nintendo.

• Terminator Hitman

  8 Sep 10, 16:27

  ~ pls do not post unrelated stuff here ~
  

• Terminator Hitman

  8 Sep 10, 16:30

  ~ pls do not post unrelated stuff here ~

• Mad Hat

  8 Sep 10, 17:37

  Originally posted by eagle:

  Hi,

  I don't think you can get the exact value with any GC if it is in terms of square roots or ln or etc type of weird non-rational values

  For TI-84, I believe you can press [2nd] [Calc], and number 5, finding intersection. The calc button is somewhere on the small blue text at the top of the calculator, around the window key I think.

  I don't own any GC, nor do I have one with me right now, so pardon me if I'm wrong.

  Thanks Eagle, we'll try that.

  Oh sorry! - doesn't have to be "exact" like root 2 or what. Decimals are fine. The problem was she was positioning the cursor by hand over the intersection point as best she could, and then noticing the co-ordinates of the cursor !

  I told her the GC must have some automatic way of declaring the intersection points but we couldn't figure it out. So I'll try [2nd] [Calc] and number 5.

  Anyone who knows for sure how to find intersection points with TI-84, please share, thanks !