Calc II-College

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notgoodatmath777

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Ray has finally saved up enough money to make his dream of opening an ice cream store come true. He measure the time t in minutes between customers arriving and tries to model it using a probability density function. He decides that the probability density functions takes the form of p(t) or r(t) where, t > 0, p(t) = a e^(-bt) and r(t) = m / (t+k)^2, where a, b, m, and k are constants.

a) How should p(t) and r(t) be defined for t < 0? Explain.

b) About 1/9 of the time, the next customer arrives within 10 seconds of the previous one. He uses this to approximate the value of his density functions at t=0. His approximation comes from the interpretation: "The probability of the next customer arriving between 0 and 10 seconds after the previous one is approximately 1/9" What should the value of his density functions at t=0 be?

c) Ray sets p(0) and r(0) equal to the value he found in (b), and solves for the values of a, b, m, and k. What values should she find? Remember that p(t) and r(t) are both probability density functions.

d) Ray finds that the median amount of time between customers arriving is slightly over 1 minute. Based on this, which of p(t) and r(t) is a better model for the arrival time between customers?
 
This is a help forum where we help students solve their own problem vs solving problems for students.
Please show us what you have done so far so we know what type of help you need. It really is hard to offer help without knowing what your thoughts are on how to work this problem.
 
Jomo said:
This is a help forum where we help students solve their own problem vs solving problems for students.
Please show us what you have done so far so we know what type of help you need. It really is hard to offer help without knowing what your thoughts are on how to work this problem.
Click to expand...

a) How would I relate this back to a pdf? I'm also unsure of what the question is asking about specifically.
p(t): As t gets smaller and smaller (more negative), the fraction of customers at time t gets larger and larger. Modeled by an exponential function.
r(t): As t gets smaller (more negative) and t<k, r(t) increases: fraction of customers at time t gets larger.

b)
p(t) at t=0: p(0) = 0 or 1?
r(t) at t=0: r(0) = 0 or 1?
If t is measured as the minutes between customers arriving and t=0, then the customers are arriving one after another.
I set the equations equal to 1/9 for t=1/6, and 0 for t=0.

c) Do I set p(t) and r(t) equal to 1/9 and set t=1/6 to solve for some of the constants? Would I take the integral to find the rest of the constants. Or set p(t) and r(t) equal to 0 and t=0?

d) Median = integral from -infinity to time t of pdf dt is equal to 1/2. Would I plug in t=1 and whichever one is closest to 1/2 but not greater than is the better equation?

Thank you, whatever help is appreciated
 
Let's look at part (a) first. Can the random variable ever be negative? What does that suggest about the PDF?

How about part (b): You are told that "About 1/9 of the time, the next customer arrives within 10 seconds of the previous one." How would you find that probability (that t is between 0 and 1/6) in terms of a PDF? It is not about the value at t=0 or t=1/6, but about ... an integral?
 
I also am stuck on this problem, but understand the concepts, just can't apply them to this problem.

This is what I have so far, please let me know how I did.

a. p(t) and r(t) cannot be negative for any value of t. If t<0, they must be zero/not defined, as time cannot be negative.

b. The integral from 0 to 1/6 of the density functions is 1/9, however, I can't get an answer, only one involving constants once I integrate and plug in the bounds. Also, I thought that p(t=0) is 0, as for a pdf, having exactly one value(in this case 0) is 0, as if you draw a graph and extend the line from 0 to the graph, the thickness is 0. If the answer is 0, than how would you find the value of the constants?

C and D are easy once finding B, appreciate any help
 
For (a), you're right that a pdf can never be negative, though that's not directly relevant here. And you're right that time here (meaning the elapsed time from one arrival to the next, is never negative, though time in other situations can be negative. But there is a difference between zero and "not defined". Can you be more precise about that? (Both answers make sense, if you explain them correctly.)

For (b), I realize now that I didn't look closely at what it asks you to do, and I was jumping to part (c). I suspect you are expected to make an approximation, supposing the t will normally be much greater than 10 seconds. What would the probability density be over this 10-second interval if it were constant, and the probability were 1/9?

I'm curious how two of you happen to be working on the same problem. Where is it from? And what have you learned about PDFs in this context?
 
For a, then I think the answer should be 0, as from negative infinity to positive infinity, the pdf must equal 1, so not defined may not be correct in this context.

For b, I'm not really sure what you are asking. Also, if you could either refute or correct my initial thoughts, that would be greatly appreciated.

Appreciate the help so far Dr.Peterson, for me at least, I have this as a homework question from University, and the textbook sections regarding pdfs do a terrible job explaining it, and the teacher doesn't really teach. The only information given about pdfs is that from -infinity to infinity they are equal to 1, and they are greater than or equal to 0 for all values
 
bub said:
For a, then I think the answer should be 0, as from negative infinity to positive infinity, the pdf must equal 1, so not defined may not be correct in this context.
Click to expand...
I agree. The "undefined" option would mean restricting the domain to non-negative numbers, which may not be considered appropriate in your context.

bub said:
For b, I'm not really sure what you are asking. Also, if you could either refute or correct my initial thoughts, that would be greatly appreciated.

Appreciate the help so far Dr.Peterson, for me at least, I have this as a homework question from University, and the textbook sections regarding pdfs do a terrible job explaining it, and the teacher doesn't really teach. The only information given about pdfs is that from -infinity to infinity they are equal to 1, and they are greater than or equal to 0 for all values
Click to expand...
My own problem is that I'm not sure how this would be taught in the context of a calculus course, so I'm not sure what to assume. I'll guess that it is being somewhat related to Riemann sums. (By the way, be careful how you word things. It is not that the pdf from [MATH]-\infty[/MATH] to [MATH]+\infty[/MATH] is "equal to 1", but that its integral is equal to 1 ...) Precision of language contributes to precision of thought!

Let's look at what you said on (b):

bub said:
b. The integral from 0 to 1/6 of the density functions is 1/9, however, I can't get an answer, only one involving constants once I integrate and plug in the bounds. Also, I thought that p(t=0) is 0, as for a pdf, having exactly one value(in this case 0) is 0, as if you draw a graph and extend the line from 0 to the graph, the thickness is 0. If the answer is 0, than how would you find the value of the constants?
Click to expand...
Neither function can be zero at t=0, as that would require a and m to be zero, which would be disastrous! I suppose what you mean by your statement about thickness being 0 is that the probability that t is [MATH]exactly [/MATH]zero will be zero, which is appropriate. This would be why the data given is not about the probability of being zero, but of being between 0 and 10 seconds. So draw your graph again, with vertical lines at t=0 and at t=1/6, and think about the area under the curve between them. If we think of this as a [MATH]y\Delta x[/MATH], or rather, as [MATH]p(t)\Delta t[/MATH], what should it equal? If we assume p(t) doesn't vary much over this very small interval, what should p(t) equal? (And likewise r(t).)
 
This is what I have, but don't know what to do from here
 

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That's what I initially thought they wanted you to do, until I read more closely and realized they first want you to find p(0).

Here's the picture I said to draw (ignore specific numbers, which I picked randomly):
1604366595165.png
If the red curve is the pdf p(t), and the blue area is P(0<X<1/6) and has to equal 1/9, what must p(0) be, approximately?
 
I have run into yet another problem. Plugging in 2/3 for t = 0, I am able to retrieve the value of a, m, and k, which are 2/3, 2, and sqrt(3) respectively. However, I am unable to get b. What would be the best way to approach this?
 
bub said:
I have run into yet another problem. Plugging in 2/3 for t = 0, I am able to retrieve the value of a, m, and k, which are 2/3, 2, and sqrt(3) respectively. However, I am unable to get b. What would be the best way to approach this?
Click to expand...
Now that I've finally actually tried to work through the entire problem, I think we have to abandon the order in which questions are asked, and just do what's right. I have no idea why the author said what they said. Skip (b) entirely.

I would first use the fact that this is a pdf to find (by means of an integral) how the two parameters in each function must be related. You'll find a simple relationship between a and b, and between m and k.

Now use the fact about 10 seconds (not the approximation I pointed out) to find (again using an integral) the values of a, b, m, and k. This gives you (c).

Now find the median in each case, and answer (d).

If this was made up by your teacher, you should by now have asked about it. I hope you'll be told that it isn't quite right.
 
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