VCE Methods Question Thread!

[TrueTears]

Btw, do you know this because you are a genius or did I miss something in class? 

It's a famous property of (irreducible and aperiodic) markov chains.

http://en.wikipedia.org/wiki/Markov_chain#Mean_recurrence_time

[joey7]

Answer confused me for this...

What is the maximal domain of f(x)=loge(|x-3|)+6

[eddybaha]

R/(3) is the maximal domain. 

[b^3]

What can we normally put into a log? Anything that is greater than zero. In this case, since we have a modulus inside the log, whatever we put in, be it negative or positive, it will become positive, with one exception. We can't have the inside of the log being zero, so that is , and so .

That is we can have any other value of besides 3, so our domain is , which makes sense if you look at the graph.
https://www.desmos.com/calculator/x16wbjilx1

NOTE: Remember for logs without the modulus you'll still have to have the inside being greater than zero.

[Flor]

Ooooo this question I like this one. I'll give you a proper explanation of the VCAA method compared to the "standard" long boring method!

From VCAA

[img width]http://i.imgur.com/OR8Ward.png[/img]

I assume from your reply that you understand ahat's method yeah, coz it's the table method in the top half of the VCAA solns.

Now, the matrix method is pretty cool, because it works on the principle for n repeats:





So, for this question you would calculate:

Thanks so much Alwin! I like the matrix method.

[joey7]

Having a bit of trouble with transformations:

Find the equation of the image of the graph of  f(x)= -2/(x-2)^2

Under a dilation of scale factor 2 from the y axis followed by
a translation of 3 units in the negative direction of the x-axis.

Express your answer in the form y=c/(x+d)^2

[ashoni]

How do you find the long term probability of something with the transitional matrix without the use of a calculator?

[eddybaha]

Having a bit of trouble with transformations:

Find the equation of the image of the graph of  f(x)= -2/(x-2)^2

Under a dilation of scale factor 2 from the y axis followed by
a translation of 3 units in the negative direction of the x-axis.

Express your answer in the form y=c/(x+d)^2

after dilating you get
y=-2/(x/2-2)^2       (x becomes x/2)
after translation
y=-2/((x+3)/2-2)^2      (x becomes x +3)
simplifying.....multiply top and bottom by 2^2
y=-8/(x-1)^2
 

[eddybaha]

How do you find the long term probability of something with the transitional matrix without the use of a calculator?

let the steady state matrix =steady state times transitional matrix
S_infinity=S_infinity*T
this is your steady state.
now solve for a and b using matrix multiplication

[sasa]

Ok, so can someone explain to me what the term 'mutually exclusive' means for prob. As in question 8, part b from the 2011 vcaa paper. I read it as (A'intersect B') but the answer says it's (A' intersect B). Em, what??

[b^3]

Mutually Exclusive implies that the probability of the intersection is zero (thinking of it as sets, the two sets have nothing in common). .
Since we want "whats not in A but in B", when there is no intersection, then that is just what's left in B, so .

[sasa]

Thanks so much!   

[zvezda]

let the steady state matrix =steady state times transitional matrix
S_infinity=S_infinity*T
(Image removed from quote.) this is your steady state.
now solve for a and b using matrix multiplication

The essentials textbook suggests otherwise??

[Damoz.G]

The essentials textbook suggests otherwise??

Steady state depends on which bit you are doing it for. But its a/(a+b) and b/(a+b)

[b^3]

Steady state depends on which bit you are doing it for. But its a/(a+b) and b/(a+b)

From memory, Essentials defines the matrix as to give the steady state probabilities that you mentioned.

If is your success and is your failure then the long run/steady state probabilities are

So you can think of it as taking the entry from the opposite diagonal that corresponds to the row for the event you want, and dividing it by the sum of the opposite diagonal.
So if you want the steady state probability of the event that you defined to be the top row of the transition matrix then you want to find .

EDIT: When I say opposite diagonal I mean the bottom left to top right diagonal.

let the steady state matrix =steady state times transitional matrix
S_infinity=S_infinity*T
(Image removed from quote.) this is your steady state.
now solve for a and b using matrix multiplication

The essentials textbook suggests otherwise??

This will probably still work, it's just a different method of looking at it and defining it.