BEC'S methods questions

[Collin Li]

But doing that for each of the sums/Y-values would be time-consuming. Is there a more efficient way of solving this?

You should do the time-consuming way. It is not that hard:

2: (1,1)
3: (1,2) or (2,1)
4: (1,3), (3,1) or (2,2)
etc.

You will see a pattern soon enough, and there should be 36 total combinations ().

There is a faster way though: it involves the binomial distribution, which is just a chapter or two after the one you're currently on.

[/0]

You learn binomial distribution in Methods 1/2 i think.

[Collin Li]

You learn binomial distribution in Methods 1/2 i think.

I don't think so, unless your teachers have been accelerating your 1&2 course to suit the 3&4 course.

http://en.wikipedia.org/wiki/Binomial_distribution

The binomial distribution wouldn't do that question at all actually. I don't know what I was thinking.

[bec]

yeah i was wondering how binomial dist would work for that one! well if the only way to do it is by working out each one i won't worry about doing it...

How about this one?

The time, T days, it takes for a seed to germinate after being planted is known to have a probability density function and zero elsewhere.
The next few questions I could answer easily enough:
a) Show that k=0.5
b) Find the probability that a seed takes longer than 1 week to germinate (=0.0302)
c) Find the probability that a seed takes longer than 1 week to germinate given that it takes longer than 3 days (=0.1353)

But then they add this bit:

If a random variable T has probability density function given by f(x) =ae ^-aet , and zero elsewhere, then it is known that and E(T²)=

d) Use the above results to find:
     i. the average time for a seed to germinate
    ii. the standard deviation of the time for a seed to germinate

e) Find the probability that the time taken for a seed to germinate is within two standard deviations of the mean.

f) Calculate the exact value of the median time for a seed to germinate.

I tried to approach it by finding the value of "a" first (nb, it's actually a greek alpha, not "a" - that doesn't mean anything special does it??)
I did it like this, without much success:

       
       
         = undefined?
so yeah, how do I do this? and what's the link between this second half of the question, and the first half?

[Ahmad]

There is not much computation if you use this method:

Consider the generating function of the die, here the power of x is the number on the die, and the coefficient is the corresponding probability (we wish to find the mean).



Now the product is the convolution of which simply gives the desired sum of the numbers on the die. Note that by taking the derivative of we are multiplying the coefficients by their value, which is simply the mean. (I'll be happy to explain more if this part didn't click).



In fact, this method is general, and for curiosities sake, the mean for 100 throws of this die is 385. (Just imagine doing this with the normal method). 

[bec]

(I'll be happy to explain more if this part didn't click).

whoa. um, you know what, don't worry about explaining...we'd both be here till next year
i'll let you know in a couple of years when it makes sense to me haha
+1 for being too smart for 17

[AppleXY]

He's physically 17 years old. But has the mind of 25 year old  (but really, anyone can do it. Just need practice I can actually understand g. functions now (a little lol) )

Ahmad is pwn tutor though. Period.

[Ahmad]

lol of course not. Anyone can do it. It's pretty simple actually, I just probably didn't explain it well.

[Collin Li]

But then they add this bit:

If a random variable T has probability density function given by , and zero elsewhere, then it is known that and

d) Use the above results to find:
     i. the average time for a seed to germinate
    ii. the standard deviation of the time for a seed to germinate

e) Find the probability that the time taken for a seed to germinate is within two standard deviations of the mean.

f) Calculate the exact value of the median time for a seed to germinate.

From the earlier parts of this question, we had this ( substituted in)


All we need to do now is recognise that in the second part, setting gives us the distribution in the earlier parts of the question.

d)
i. the average time for a seed to germinate
Using , and by recognition, using (to yield ), then we get:



ii. the standard deviation of the time for a seed to germinate
Applying same principle as before ():






e)
2 standard deviations from the mean: (the mean is 2, and the standard deviation is also 2)
Lower bound = 2 - 2(2) = -2
Upper bound = 2 + 2(2) = 6

However, the domain of the function is for . This means we have to integrate from 0 to 6, and then -2 to 0 for (which will just give us zero).

My answer: (it's a very simple integral, if you need help getting here, just ask)


f)
Integrate from 0 to m, and let that integral be equal to 0.5. Find the value of m.

My answer: (arriving here just uses the same integral above with a bit of log laws)

[bec]

coblin you are my hero.

does anyone have a ti-89? (the cas calculator)
i bought mine second hand and i've just realised it doesn't have something i need to calculate normal probabilities. they are:
"CATALOG F3 FlashApps" and the "Stats/List Editor App".

can you get these things transferred over? or is there a way of calculating the more complex normal probabilities without those programs? eg, Q12-16 in Ch13 review in mathworld - i can't work out how to do them.

ps. in that question that's just been solved about germinating seeds, the second "f(t)" was actually denoted as "f(x)" in the book. why would they do that, when it was actually a "function in terms of t" with no x's in sight? just a mistake in the text or does it mean something?

[Collin Li]

Yes, it's a mistake. They should have had it as f(x), and replaced the t and T's with x and X, or just made it all T.

[Matt The Rat]

Bec, you can get the Stats/List Editor sent over from one calc to another. I had it sent to mine this year for Methods.

Not sure if you can calculate it without the program, but I think that a genius kid in my class (50 raw and he was in Year 11) worked out how to do it by hand.

[bec]

awesome, i was a bit worried i'd have to fork out to buy another one. whoa someone got 50 in year 11?? how is that possible!?

The times taken for a large group of swimmers to swim 50 metres freestyle are normally distributed with mean 23.1 seconds and variance 0.16 seconds.

If one of these swimmers is randomly chosen, what is the probability that they can swim 50 metres freestyle in less than 22.3 seconds?

I did it a couple of times and got 0.025 both times. The text says it’s How do you work it out?

Find the percentage of these swimmers who cannot swim 50 metres freestyle in less than 23.7 seconds.
I keep getting 9.25%, the solution in the book is 1.328.

or....are these more things that i can only do with the Stats/List app?

thanksss

[Collin Li]

I concur with your answer. I have no idea how they get a fraction so specific for a normal distribution question. Not to mention, that fraction is larger than 1. Probabilities can't be greater than 1, LOL.

Find the percentage of these swimmers who cannot swim 50 metres freestyle in less than 23.7 seconds.

[normalcdf() is a TI-83 function, there is probably a TI-89 equivalent]

I don't concur with either of the answers. I got 6.68%

Are you sure you're not looking at the wrong part of the answers?

[bec]

hahaha i didn't even think of that (the probability being over 1)! this book has so many mistakes! but the explanations are so good it makes up for it, except when there's a (vital) mistake in the explanation (ie, getting the formula for E(X) wrong in the summary which meant i couldn't answer half the questions!)

that said, i think fsn people should collaborate and write a book, it'd be better!

(ok for the second part of my question i think the thing i need to work it out is the program on my calc that i don't have - i'll learn it next year... but thanks anyway)