VCE Methods Question Thread!

[polar1]

Hi all,

I need help with this probability density function question, where f(x) = 6e^-6x, x= time in hours, and I want to find the average waiting time in minutes.

In the textbook example it states that for this random variable X, E(X)= 1/alpha and E(X^2)= 2/alpha^2 ... I'm slightly confused with this concept, so could anyone help explain this to me? Couldn't we just integrate and multiply the probability density function by x and solve?

Thank you!! 

Also, it's definitely worth trying to do it by hand. You'll need to use integration by recognition, but aside from that, it's not too bad.

If anyone wants to try it out

(a) Differentiate the function .
(b) Verify that the function can be a PDF.
(c) Use the above two results to find the expected value of a random variable with PDF . (This might be a bit tricky, you'll need to know that the a function like goes to much faster than a function like goes to infinity).

[#J.Procrastinator]

For the exponential distribution , it has mean and variance , which is what they've pulled out. These definitions can easily be proved via integration (as you've suggested doing), however note that exponential distributions aren't covered in methods, and so you do not need to memorise all of this. If asked about this stuff, I would simply do it as you would any other PDF.

Also, it's definitely worth trying to do it by hand. You'll need to use integration by recognition, but aside from that, it's not too bad.

If anyone wants to try it out

(a) Differentiate the function .
(b) Verify that the function can be a PDF.
(c) Use the above two results to find the expected value of a random variable with PDF . (This might be a bit tricky, you'll need to know that the a function like goes to much faster than a function like goes to infinity).

Oh I see, thanks

Also why is it that when we find the probability of, let's say x= 30, for a particular probability density function, it yields 0? I understand that when we integrate at that point it can't possibly have an area. Hence, the 0 outcome.

For example, let's say shoe sizes can take values of [20,65], why is the probability of x=35, 0?
Wouldn't it be quite possible for someone to have a 35cm shoe size? I don't know if I'm making sense haha :s or I'm probably missing something here .. Thanks!

[#J.Procrastinator]

Oh! and one last thing... maybe  How exactly do we show that a function is a probability density function for f(x) ≥ 0 for all real numbers of x? Do we show this by graphing the function or is there another way?

[polar1]

Oh! and one last thing... maybe  How exactly do we show that a function is a probability density function for f(x) ≥ 0 for all real numbers of x? Do we show this by graphing the function or is there another way?

Think about what it means for in terms of the range.

If you think about it that way, it's just a Methods 1&2 problem of finding the range given the domain. After you get the range, you just need to check whether it's a subset of the non-negative numbers.

Sorry! Didn't see the other question before. There's a few ways you can think about it. I'll give you two.
Firstly, if you recall how you actually got an integral, you had to take rectangles and add up all their areas. So, you had something like for the area of each strip. Even in the integral, you were never adding up single values, you were clumping values together in intervals and then adding up the rectangles they were a part of. Thus, it doesn't make sense that when you integrate a PDF on a single value, you should end up with a non-zero number.

Another way to think about it is that it is so unlikely to occur. Let us choose a person and measure their foot size. Let's say that we determined it with complete accuracy. Now, choose a million, a billion, etc. people and measure their foot sizes, how many of them will have exactly the same foot size as the first person you chose? Probably 0.

You can foot size as a string of numbers, and each digit has to be exactly the same for a position for two numbers to be the same. There are ten digits to choose from, so even assuming it was uniform, there are 10 choices for the first digit, then another 10 for the second digit, that's a hundred, and, another 10,....

[keltingmeith]

Oh! and one last thing... maybe  How exactly do we show that a function is a probability density function for f(x) ≥ 0 for all real numbers of x? Do we show this by graphing the function or is there another way?

I personally believe that the easiest way is to do it by graphing, but you can also do it by algebra.

For example, consider the graph of . If , then , so but be nonnegative for all values of x.

[TrueTears]

Hi all,

I need help with this probability density function question, where f(x) = 6e^-6x, x= time in hours, and I want to find the average waiting time in minutes.

In the textbook example it states that for this random variable X, E(X)= 1/alpha and E(X^2)= 2/alpha^2 ... I'm slightly confused with this concept, so could anyone help explain this to me? Couldn't we just integrate and multiply the probability density function by x and solve?

Thank you!! 

The waiting time of a homogenous Poisson process is defined to be exponentially distributed (iid) [http://en.wikipedia.org/wiki/Poisson_process#Definition]

Oh I see, thanks

Also why is it that when we find the probability of, let's say x= 30, for a particular probability density function, it yields 0? I understand that when we integrate at that point it can't possibly have an area. Hence, the 0 outcome.

For example, let's say shoe sizes can take values of [20,65], why is the probability of x=35, 0?
Wouldn't it be quite possible for someone to have a 35cm shoe size? I don't know if I'm making sense haha :s or I'm probably missing something here .. Thanks!

If you want to be really rigorous, the idea is because for any continuous random variable, any particular occurrence has 0 atom (i.e., 'atomless'). [http://en.wikipedia.org/wiki/Atom_%28measure_theory%29]

[LiquidPaperz]

I've got 1 quick question and a normal question

for Q12) for part d it asks if its increasing more rapidly, i know that its increasing but i tried to see if it was increasing more rapidly so i did when r=10 and r = 11, minused these two and got 6.2831.. (formula is dA/dt = 2pi r) and then i compared r= 30 and r = 31 and got 6.2831.. hence it is not increasing more rapidly but answer says it is?

For Q14, how do i do part a and b? how can you have a rectangular fish tank has a square base? 

https://drive.google.com/folderview?id=0B3uruRCX0QlQOGlHb2RhSHR1WHM&usp=sharing

thanks

[Phy124]

I've got 1 quick question and a normal question

for Q12) for part d it asks if its increasing more rapidly, i know that its increasing but i tried to see if it was increasing more rapidly so i did when r=10 and r = 11, minused these two and got 6.2831.. (formula is dA/dt = 2pi r) and then i compared r= 30 and r = 31 and got 6.2831.. hence it is not increasing more rapidly but answer says it is?

The rate at which the area increases with respect to time is dA/dt and we know this has a linear relationship with r as dA/dt = 2pir. So, as r increases the rate at which the area increases will increase in turn and hence the area will be increasing more rapidly.

For Q14, how do i do part a and b? how can you have a rectangular fish tank has a square base? 

https://drive.google.com/folderview?id=0B3uruRCX0QlQOGlHb2RhSHR1WHM&usp=sharing

thanks

A square is a type of rectangle. If the rectangle is square then we have the special case where (for the base) w = l. We know that the height is half the base length so h = l/2 = w/2, or otherwise l = w = 2h. Therefore V = lwh = (2h)(2h)(h) = 4h³.

[bobisnotmyname]

just a quick question. how do you antidifferentiate x*ln(x) and any other multiplications. or do we get to use calulator

[keltingmeith]

just a quick question. how do you antidifferentiate x*ln(x) and any other multiplications. or do we get to use calulator

Integration by parts/recognition. The first isn't covered in methods, the second they need to lead you in with. Eg, differentiate "



Otherwise, you just use your CAS.

[bobisnotmyname]

Integration by parts/recognition. The first isn't covered in methods, the second they need to lead you in with. Eg, differentiate "



Otherwise, you just use your CAS.

thanks for your help

[psyxwar]

From the ATARnotes methods book, apparently E(X) of a transition matrix for n iterations is T^0*S0 + T^1*S0 + T^2*S0... + T^n*S0. Is this on the course/ are we allowed to use it?

[Jason12]

what does the normal cdf mean and what kinds of questions would it be used for?

e.g. if you have a normal pdf and they with a mean of 5 and standard deviation of 3 and they want the percentage above 6

[Reus]

Methdos theory = done.
Exam preparation here I come.

[psyxwar]

what does the normal cdf mean and what kinds of questions would it be used for?

e.g. if you have a normal pdf and they with a mean of 5 and standard deviation of 3 and they want the percentage above 6

normal cdf and inverse normal are the only 2 normal functions on the CAS you need/ that are relevant for VCE.

Normal CDF let's you find the probability between two points on a normal curve with mean μ and standard deviation σ. The C stands for culminative, and just means sum (of the probabilities between the points). Don't worry about normalPDF, it's not relevant to VCE

For the question you gave, you'd go to the normalCDF screen (menu->5->5->2 on TI NSpire CX CAS), and type in the lower and upper bounds (so in your example, the lower bound is 6 and the upper bound is infinity, as you want Pr(X>6) which means X must be between 6 and infinity), mean as 5 and standard deviation as 3. Then you just press enter and get an answer.