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Old 20th March 2012, 03:12 AM   #1 (permalink)
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need some help with probabilities (might need some explanation)

Let X be the number of years a particular type of machine is in good working condition
and need no replacement. Assume that X has the probability mass function
P(1) = 0.4, P(2) = 0.3, P(3) = 0.2, P(4) = 0.08, P(5) = 0.02.
Find the distribution function F. Sketch F.
What is the probability that the machine needs no replacement during the first 3 years.
i was practicing my probability question when i met this 1st question and straight away i was unable to do it. the answer is 0.3. why?

i realli WTF-ed myself. did i misinterpret the question? shouldn't it be 0.4+0.3+0.2?


seriously? answer = 0.84? i dun need the working for this question as they have already showed me, but y? y use bayes theorem?


honestly speaking for the chapter of probabilities, when my lecturer goes thru them, it seems easy and a piece of cake. but the problem i am facing is that she didn't teach and tell mi how to identify them (too little samples are shown to us, some samples that she showed us are even repeated)!!! can somebody tell mi how do i identify what theorem to use for what question?

for example: at times, when they are calculating probabilities, they add while at times, they multiply.
what is this bayes's theorem, and all the other theorem, when should i apply them? are thr any special methods to use to identify this type of question?

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Old 20th March 2012, 03:38 AM   #2 (permalink)
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Re: need some help with probabilities (might need some explanation)

some theory to ask. this is from my understanding. so plz correct mi if i am wrong here.

example:
MC = machine set up correctly
GP = good product

P(GP|MC) = if written in words = getting a good product AFTER setting up the machine correctly.
the word "AFTER" is being represented by the " | ".

therefore, P(MC | GP) = setting up the machine correctly AFTER getting a good product. (sounds logically wrong, but just accept it 1st?)

would the values of MC|GP and GP|MC be the same?
also,
given that P(MC) = 0.7
P(GP | MC) = 0.9

am i able to find the GP if just given this 2 value? if not, what am i suppose to have in order to find GP?
my logic is totally wrong, but still i am going to type it out (u might laugh)
P(GP) * P(MC) = 0.9
P(GP) * 0.7 = 0.9
P(GP) = 1.29 (LOL, i know the answer is totally out of the world, but i am just showing u my initial thoughts on how i tot it was).

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Old 20th March 2012, 07:15 AM   #3 (permalink)
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Re: need some help with probabilities (might need some explanation)

maybe these words will help better. "|" represents given, known.

P(GP|MC) = if written in words = probability of getting a good product GIVEN/KNOWING THAT the machine is set up correctly.

P(MC|GP) = probability of the machine setting up correctly GIVEN/KNOWING THAT you got a good product.

no. the values will not be the same.
P(A & B) = P(B & A)
P(A | B) =/= P(B | A)

you learn to use Bayes theorem when you have a conditional probability and realize you have known values for the flip. eg. solve P(A|B) but you have values for P(X|A)

given that P(MC) = 0.7
P(GP | MC) = 0.9

am i able to find the GP if just given this 2 value? if not, what am i suppose to have in order to find GP?
unless they are independent, i don't think you can. you are missing P(MC | GP) or P(GP | MC not). and from the question, you will see that P(GP | MC not) = 0.4. after you obtained P(GP | MC not), you can use the law of total probability.

i realli WTF-ed myself. did i misinterpret the question? shouldn't it be 0.4+0.3+0.2?
you are adding up the probability that the machine will break down after 1, 2 and 3 years respectively. since question is asking that it won't break down in first 3 years, you expect the machine to either break down after 3 years or more. this means, count from the back instead.
P(5) will not break down before 3 years right?
P(4) will not break down before 3 years right?
P(3) will not break down before 3 years right?
P(2) will not break down before 2 years but may break down before 3 years right?
P(1) will not break down before 1 years but may break down before 3 years right?

so P(5) + P(4) + P(3) = 0.3

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Old 2nd April 2012, 12:15 AM   #4 (permalink)
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Re: need some help with probabilities (might need some explanation)

i think i am in confusion again.

firstly:
sometimes for standard deviation they sign that they use is "sigma" while sometimes they would use "S".
sometimes for variance they use "sigma sq" while sometimes they would use "S^2".

is there a difference? if there is when would i know when i should use what?

secondly,
what is this chi-square thing. when exactly do i use it? i am so confuse.

thirdly,
is there a difference between standard normal and normal deviation?

last but not least.
y is it that in some case, when i am looking for "z", i have to +/0.5 before looking at the probability table (the bell shape curve)


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Old 2nd April 2012, 12:41 AM   #5 (permalink)
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Re: need some help with probabilities (might need some explanation)

firstly:
sometimes for standard deviation they sign that they use is "sigma" while sometimes they would use "S".
sometimes for variance they use "sigma sq" while sometimes they would use "S^2".

is there a difference? if there is when would i know when i should use what?

Sigma is typically used to denote population standard deviation, while s is used to denote the sample standard deviation.


secondly,
what is this chi-square thing. when exactly do i use it? i am so confuse.

Imagine you have to conduct a certain experiment. Beforehand you have a list of expected(theoretical) results. Then upon conclusion of the experiment you have a corresponding list of measured (actual) results. Chi Square test is employed to investigate the goodness of fit between expected and actual result sets.


thirdly,
is there a difference between standard normal and normal deviation?

Standard normal deviation MUST have a value of 1, while normal deviation can assume any value.

last but not least.
y is it that in some case, when i am looking for "z", i have to +/0.5 before looking at the probability table (the bell shape curve)

Are you talking about continuity correction? This happens when you are approximating discrete distributions to continuous distributions (eg binomial to normal).

Hope this helps. Peace.


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Old 2nd April 2012, 01:16 AM   #6 (permalink)
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Re: need some help with probabilities (might need some explanation)

last but not least.
y is it that in some case, when i am looking for "z", i have to +/0.5 before looking at the probability table (the bell shape curve)

Are you talking about continuity correction? This happens when you are approximating discrete distributions to continuous distributions (eg binomial to normal).

Hope this helps. Peace.
maybe i didn't phrase it properly. i am not sure if what u have mention is similar but anyway, i was refering to:
when do we let X = normally distributed and X = approximately normally distributed?

care to elaborate more on these terms? coz i dun realli see the pt why we let X be approximately normal while sometimes not. the working for these 2 are different as they would like x be +/- 0.5 when x is in between 2 values? e,g (-2<x<5), den they would let it be (-2.5<x<5.5)....why? what if i just let the value be itself w/o the +/- 0.5?

PS: thanks for explaining.


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Old 2nd April 2012, 01:26 AM   #7 (permalink)
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Re: need some help with probabilities (might need some explanation)

sry. but i need to ask again, for the 1st qn, does it make a difference if i use sigma to denote a sample standard deviation? since the formulas are the same rite?

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Old 2nd April 2012, 01:28 AM   #8 (permalink)
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Re: need some help with probabilities (might need some explanation)

Originally Posted by KiLlErDeViL View Post
maybe i didn't phrase it properly. i am not sure if what u have mention is similar but anyway, i was refering to:
when do we let X = normally distributed and X = approximately normally distributed?

care to elaborate more on these terms? coz i dun realli see the pt why we let X be approximately normal while sometimes not. the working for these 2 are different as they would like x be +/- 0.5 when x is in between 2 values? e,g (-2<x<5), den they would let it be (-2.5<x<5.5)....why? what if i just let the value be itself w/o the +/- 0.5?

PS: thanks for explaining the 1st three questions.
If it is stated explicitly within the question itself that things are normally distributed, then X is exactly =normally distributed. Sometimes (for example) you encounter X which is an exact binomial random variable, but when certain conditions are met, then X can be approximately transformed into one which is modelled by the normal distribution.

Note: you may wish to take a look at Q9 on this page of my site:
http://www.whitegroupmaths.com/2010/...nd-normal.html

Continuity correction (ie the +/-0.5 thingy) is meant to compensate for approximation errors(call it bridging the gaps if you must) when migrating to a continuous distribution platform from a previously discrete one. If you don't do this, the accuracy of your answers might be significantly diminished.

PS: (-2<x<5) after cc is (-1.5 < x < 4.5); it is (-2<= x <= 5) which would give you ( -2.5, 5.5) . The inequality structure matters.

[ <= means lesser or equals to]

Hope this helps. Peace.

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Old 2nd April 2012, 01:31 AM   #9 (permalink)
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Re: need some help with probabilities (might need some explanation)

Originally Posted by KiLlErDeViL View Post
sry. but i need to ask again, for the 1st qn, does it make a difference if i use sigma to denote a sample standard deviation? since the formulas are the same rite?
Sigma is usually given to you. The formula given in most notes refer to the computation of sample standard deviation.

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Old 2nd April 2012, 01:36 AM   #10 (permalink)
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Re: need some help with probabilities (might need some explanation)

i think i get it, but still abit lost. will get back to you again when i figure out what is really confusing me again.


as always, thanks for helping.

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Old 2nd April 2012, 01:38 AM   #11 (permalink)
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Re: need some help with probabilities (might need some explanation)

Ok I am not going to deny this slightly ambiguous representation of standard deviation in various notes and books:

Sigma refers to the population standard deviation; sigma square refers to the population variance. This part is fine.

Now the troublesome part.

These days, JCs use s^2 to denote unbiased estimate of population variance, which is slightly different from that of the sample variance. If sample variance is a certain value say A, then JC students are required to state that s^2 is n/(n-1) times A, where n is the sample size. Yet back then, I was taught that sigma (hat) ^2 represents the unbiased estimate of the population variance, while s^2 denotes the sample variance. (why do you think they call it s? Because s stands for Sample)

Since you are in university, I won't want to make your brains bleed, so take s^2 as sample variance. End of story.

Hope this helps. Peace.


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Old 5th April 2012, 02:08 AM   #12 (permalink)
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Re: need some help with probabilities (might need some explanation)

in this topic there are so many "common terms" that have made my life really miserable. now i am a little lost at the meaning of "mean" and "average". aren't they the same?

for e.g. for the below 2 qn:
a new brand of canned mixed nuts plans to enter the singapore market. the label says that 25% of the contents are cashews. suspecting that this might be an overstatements, an inspector takes a random sample of 31 cans and measures the eprcentage weight of cashews in each can. the mean and standard deviation of these measurements are found to be 23.5 and 3.1, respectively. do these results constitute strong evidence in support of the inspectors' belief...........
timesless is a company tat manufactures quartz crystal watches. timeless researchers have shown that the watches have an average life of 28mths before certain electronics components deteriorate, causing the watch to become unreliable. the standard deviation of watch lifetimes is 5mths, and the distribution of lifetimes is normal .........
for the above 2 qn, 1 use the word "mean" while the other use "average". is there a difference? if they use the term "mean" does it mean that i have to use x(bar) which represents sample mean, while the term "average" would means that i have to use the "mew (the weird sign)"? the 1st qn also used the 25% as the "mew". y?



also previously i asked abt the normally distributed and approximately normally distributed (which had their own formulas). i questioned my lecturer abt it, he told me that as long as the sample size "n" is more than 30, use approximately normally distributed. else use normally distributed.
- to be honest, i am not really satisfied with the way he answered it. coz there is no logic to it.


sigh... how did math ever turn into a english test


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Old 6th April 2012, 12:14 AM   #13 (permalink)
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Re: need some help with probabilities (might need some explanation)

There is no difference between the words mean and average (they can be used interchangeably); however what matters is that you clearly identify within the question itself what exactly is deemed as the population mean (symbol mu) and sample mean (symbol x bar). For instance, in the first question, the population mean is 25 (which you have to conduct a hypothesis test to ascertain if the value has been overstated) while the sample mean is 23.5. You will using the sample mean and sample variance to conduct the hypothesis testing through the evaluation of a relevant test statistic.

Try and see if you can identify mu and x bar in the second question (you did not post up the entire question,ie certain numbers are absent, so I can't give you complete advice-still it shouldn't be difficult to figure things out).

Regarding all things tending towards a normal distribution when the sample size is sufficiently large, this is based on the concept known as central limit theorem. It is hard to articulate it fully to you over the web, so you will have to do some reading up on your own. But seriously, shouldn't you know all these stuff before you even entered university? It is really elementary, basic statistics material, in fact all the above is only of JC H2 maths standard.

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Old 6th April 2012, 01:03 AM   #14 (permalink)
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Re: need some help with probabilities (might need some explanation)

poly doesnt teach such stuff

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Old 6th April 2012, 01:06 AM   #15 (permalink)
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Re: need some help with probabilities (might need some explanation)

Oh my god. No wonder you sound so lost.

Anyways if you have any other queries feel free to post them. I will try my best to sort them out when I am available. Peace.

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Old 6th April 2012, 11:08 AM   #16 (permalink)
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Re: need some help with probabilities (might need some explanation)

not sure if it would be appropriate but i dun wan to create too many thread, so i think i will be posting some other math qn tat is not related to probability here.


the way i phrase my qn may be hard to understand. but since i am still in sch, and do not have a scanner, i cannot scan my working in. if u do not understand what am i saying, i will try to scan in my working by sat afternoon when i go home.


when attempting this qn, i managed to get till
Uc = A exp (sx) + B exp (-sx)
Up = ???
where U = Uc + Up

solution that was given to me, they let
Up = C cos (pi x) + D sin (pi x)

my only qn would be how they get Up = the above equation? i always tot that Up is fixed and is always Up = Ax + B. afterwhich i just nid to Up' and Up''.

PS: good friday to you. hope u enjoy ur long weekend.


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Old 8th April 2012, 01:33 PM   #17 (permalink)
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Re: need some help with probabilities (might need some explanation)

Originally Posted by KiLlErDeViL View Post

my only qn would be how they get Up = the above equation? i always tot that Up is fixed and is always Up = Ax + B. afterwhich i just nid to Up' and Up''.
That is incorrect. You formulate your Up based on the outlook of the distinct function (in x) not tied together with U, U' and U", ie in this case -sin (pi x). So Your Up must be of a trigonometric form as well.

I have worked the part you are concerned with in full, please take a look below. Hope it helps. Peace.



Note: For (1) in the image above, unless the RHS is presented along the lines of for example 11x+3 (a linear function in x), then we must let Up be equal to Cx+D.


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Old 8th April 2012, 02:09 PM   #18 (permalink)
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Re: need some help with probabilities (might need some explanation)

hmm. in other words, if RHS is a linear function, i can let Up = Cx + D and
if it is a trigo, i let Up = C cos x + D sin x. (without the pi)
den what abt if RHS is a non-linear function or an exponential function?

i flipped thru my lecture notes, and they only teach linear function. zzz.. FYI, the above qn is a past year paper qn (might seem easy for u). it realli sux to have this type of qn when it wasn't even taught.


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Old 8th April 2012, 02:21 PM   #19 (permalink)
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Re: need some help with probabilities (might need some explanation)

Originally Posted by KiLlErDeViL View Post
hmm. in other words, if RHS is a linear function, i can let Up = Cx + D and
if it is a trigo, i let Up = C cos x + D sin x. (without the pi)
den what abt if RHS is a non-linear function or an exponential function?

i flipped thru my lecture notes, and they only teach linear function. zzz.. FYI, the above qn is a past year paper qn (might seem easy for u). it realli sux to have this type of qn when it wasn't even taught.
Yes, you have to look at the RHS. If RHS = say 3 cos2x, then we let Up = C cos2x+D sin2x; if RHS = 3cos5x, then we let Up =C cos 5x +D sin 5x.

If RHS= say 5x^2 + 2x -3 (ie non-linear), then we let Up= Cx^2 +Dx +E
If RHS =say 7e^(12x), then we let Up=Ce^(12x)

Note that the Laplace Transform we apply to the second order partial DE transforms it into a second order ordinary DE, so we will solve things in the ordinary DE manner subsequently.

If you want, I can complete the solution for you (would be great if you have an end answer to check against).

Hope this helps. Peace.

(There will come a time when the degree of difficulty of your stuff exceeds my ability, which means you have to seek out other sources of assistance. )


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Old 8th April 2012, 02:43 PM   #20 (permalink)
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Re: need some help with probabilities (might need some explanation)

Originally Posted by whitecorp View Post
Yes, you have to look at the RHS. If RHS = say 3 cos2x, then we let Up = C cos2x+D sin2x; if RHS = 3cos5x, then we let Up =C cos 5x +D sin 5x.

If RHS= say 5x^2 + 2x -3 (ie non-linear), then we let Up= Cx^2 +Dx +E
If RHS =say 7e^(12x), then we let Up=Ce^(12x)

Note that the Laplace Transform we apply to the second order partial DE transforms it into a second order ordinary DE, so we will solve things in the ordinary DE manner subsequently.

If you want, I can complete the solution for you (would be great if you have an end answer to check against).

Hope this helps. Peace.

(There will come a time when the degree of difficulty of your stuff exceeds my ability, which means you have to seek out other sources of assistance. )
thx alot, but u dun have to trouble urself to get the soln out. i will work things out here myself.


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