Question

The continuous random variable X has distribution function given by F(x)=1-e^(-1/2 * x)
Find the probability that 2 lies between X and 4X,
FInd also the expected value of the width of the interval ( X,4X)

Answer 1

\[F(x)=1-e^{-\frac{ 1 }{ 2 }x} (x \ge 0)\]

Answer 2

Okay, I think the first step would be to find the probability density function (PDF). Are you familiar with how to do that?

Answer 3

No! I was sleeping during my STAT class, that's why I come here

Answer 4

To find the probability density function, I believe you can simply take the derivative of the the distribution function, F(x).

Answer 5

So derivative of F(x) = F ' (x) = ...?

Answer 6

WTF is distribution fuction? Is it short for Cumulative Distribution Function?

Answer 7

Yes, a continuous distribution function (cdf) is what a distribution function is referring to.

Answer 8

well, dF(x)/dt = 1/2*e^(-1/2 * x)

Answer 9

http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/fstats_ch1.pdf

Answer 10

OMG! Where do you find it?! What a wonderful Xmas gift!

Answer 11

BTW, are you studying further math now?

Answer 12

I typed "Find the probability that 2 lies between X and 4X" into google and it was one of the first things that came up.

Answer 13

LOL! Google knows what I need!

Answer 14

I have a bachelor's degree in math. I am not formally studying math at the moment, but I do enjoy helping others learn math (and learning with them).

Answer 15

Why the CDF of a poisson distribution is given as 1-P(X=0) Where P(x=0) means the probability of obtaining o when a random variable X follows a poisson distribution of parameter lambda

Answer 16

Where does it say that the poisson distribution is 1-P(X=0)? What page number or weblink?

Answer 17

http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20-%20Further%20(9231)/9231_s08_ms_2.pdf

Page 6, Question8

Answer 18

@Yttrium , I'm goinf to be killed by this question!

Answer 19

I wonder why you guys are doing maths on Christmas day of all days...

Answer 20

@terenzreignz , both my math and physics teachers gave me a bunch of questions as "Christmas Gift"! WTF!!!!

Answer 21

I'm here, there is no need to tag me.
I'm just here to watch you suffer >:)

Answer 22

@Yttrium , you should find solution here; http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20-%20Further%20(9231)/9231_s10_ms_21.pdf
It is question 7

Answer 23

I hate you terenz!

Answer 24

You do?
But didn't I save your life at least twice a while back?
:-?

Answer 25

Yes, you did , but you are not so kind today

Answer 26

And thus you hate me? :(
Oh well, then that's not much incentive for me to step up, is it? :D

Answer 27

I mean " Watch me suffer" is not an appropriate Xmas gift, is it?

Answer 28

awww lighten up, it's Christmas... and besides, can't you take a little humour? :3

Answer 29

I'sorry, maybe Chinese does not have abundant humor

Answer 30

Racist >:)
LOL JK
I'm only half-Chinese.

What's the PDF?

Answer 31

http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20-%20Further%20(9231)/9231_s10_ms_21.pdf

Answer 32

all CIE subjects suck

Answer 33

In particular further math

Answer 34

I cannot see your question in that link.

Answer 35

It should be question 7

Answer 36

That pdf is the answer part

Answer 37

Is this the answer key? D:

Answer 38

if you want the question PDF, change ms at the end into qp

Answer 39

ohhh....
When I said PDF, I meant probability density function -__

Answer 40

LOL! that is so confusing!

Answer 41

Anyway, as stated, your probability density function is the derivative of your distribution function... though, in this question, it's not necessary to differentiate...

Answer 42

http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20-%20Further%20(9231)/9231_s10_qp_21.pdf

have a look at question 7 here

Answer 43

I don't understand the CIE MS, it's just a mess

Answer 44

Listen *to ME*
>:)

The only reason we get the PDF is so that we can integrate it to find probabilities... but if we're going to integrate the PDF (which is the derivative of your distribution function F(x) )
We'll arrive back at F(x) anyway, so let's skip that part.

Understand?

Answer 45

I'm still a little bit confused on the concept of CDF, what does it represent?

Answer 46

If F(x) is the CDF, it means F(c) is the probability that x < c

Answer 47

My stat's teacher's lecture drive me to sleep, so I did not hear any words he said.......

Answer 48

ok, that makes sense

Answer 49

so PDF is the probability of x=c, right?

Answer 50

Such an un-dilligent student >:)
...LOL moving on.

So, if we let f(x) = F'(x) be the pdf, then the probability that a < x < b is given simply by
\[\Large \int\limits_{a}^bf(x) dx=\int\limits_a^bF'(x)dx = F(b)-F(a)\] right?

Answer 51

Yes, I remember that from my Pure Class when my teacher talked about Fundamental Theorem of Calculus

Answer 52

Well then, what we want to find is this:
\[\Large x < 2 < 4x\]
so this looks... hard, doesn't it?

Answer 53

Because what we DO know how to solve is something of the form
a < x < b
But not if x is in the boundaries... so what to do?

Answer 54

I don'tknow!

Answer 55

well, shall we just sit here and wait for a miracle? XD

Answer 56

Maybe we should create one

Answer 57

Sure... go ahead, make a miracle happen :D

Answer 58

I can't , I'm not magician

Answer 59

Me neither... so what we do? D:

Answer 60

Mark Scheme is Powerful!

Answer 61

Who/what is a Mark Scheme?

Answer 62

But it does not help when I cannot understand what it is talking about

Answer 63

it is the following:
http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20-%20Further%20(9231)/9231_s10_ms_21.pdf

Answer 64

LOL
You mean the answer key?
I call it spoilers...
it takes the fun out of stuff.
And anyway, I already have worked my magic, I was just hoping you'd be able to do it yourself without my interference :D

Answer 65

I'm pretty sure we will have some superposition, as if we are waves, LOL!

Answer 66

Silence. >:(

Answer 67

Let's find a spoiler then

Answer 68

Now, pupil, we have
\[\Large x < 2 < 4x\]
Suppose we divide everything by x (which is a positive random value)
\[\Large 1 < \frac2x < 4\]
And now, get the reciprocal of all sides... (which will invert the inequalities)
\[\Large 1 > \frac{x}2>\frac14\]And finally, multiply everything by 2, we get
\[\Large \frac12 < x < 2\]


Much like last time, you're sorely lacking in creativity.... you disappoint me >:(


(LOL ^_^)

Answer 69

but the question says \[x \ge 0\]
I don't think we should ignore the case when x=0

Answer 70

Of course we shouldn't. But x = 0 already doesn't fit the criterion because, clearly, when
x = 0
then it is NOT true
that x < 2 < 4x

Answer 71

OMG! you're the Magician!

Answer 72

A spoiler tells me that E(x)=2, but why?

Answer 73

From now on, you will address me as MASTER Terenz >:)

LOL

Now, all you have to do is integrate the pdf from 0.5 to 2.
\[\Large \int\limits_{\frac12}^2f(x)dx = F(2) -F\left(\frac12\right) \]

Answer 74

Ok, that's kindergarden stuff

Answer 75

Show-off >:(
Anyway, to get the expectations, this time, you DO need the pdf in its explicit form, so go ahead, differentiate the distribution function F(x)
F'(x) = ?

Answer 76

it is\[\frac{ 1 }{ 2}\times e ^{-\frac{ 1 }{ 2 }x}\]

Answer 77

Okay, now, \[\Large f(x) = \frac12e^{-\frac12x}\]

Find
\[\Large \mathbb{E}[x]=\int\limits_0^\infty xf(x) dx\]

Answer 78

Should we use "by parts" this time?

Answer 79

I thought this was kindergarten stuff, pupil? >:)

Answer 80

No, it's high school stuff

Answer 81

Then it's easy stuff. Integrate it in the manner that seems most appealing to you... (but make sure it's correct :D )

Answer 82

for a second there i thought you were talking bout the .pdf file, then i scrolled up to learn a new math term "probability density function " lol

Answer 83

We learn something everyday... even on Christmas day :D

Answer 84

My Christmas present to myself is catching up on missing assignments ( though i don't celebrate it ). What a wonderful gift, right? :D

Answer 85

Time will tell :D

Answer 86

lol

Answer 87

OK, now I see E(x) is 2

Answer 88

but by parts is so tiring, is there a easy way

Answer 89

Nope... are you telling me you are afraid of working hard, my pupil? D:

Answer 90

I'm afraid of Integration By Parts!

Answer 91

I was knocked down. -____-"

Answer 92

Conquer your fears.
Now, a property of E(x) is that it's linear.. IE
E(x + y) = E(x) + E(y)
and
E(cx) = cE(x) for some constant c.

NOW
What is the length of the interval [x,4x] ?

Answer 93

what does the term " width of the interval" mean, is it just 3X

Answer 94

Yes it is.
Now you need to find the expected value of the LENGTH.
Which is 3x.
So...?
E(3x) = ?

Answer 95

oh sorry, width, not length.

Answer 96

ok, now I see

Answer 97

So... everything is settled, then?