GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 17 Nov 2018, 05:54

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel
Events & Promotions in November
PrevNext
SuMoTuWeThFrSa
28293031123
45678910
11121314151617
18192021222324
2526272829301
Open Detailed Calendar
  • FREE Quant Workshop by e-GMAT!

     November 18, 2018

     November 18, 2018

     07:00 AM PST

     09:00 AM PST

    Get personalized insights on how to achieve your Target Quant Score. November 18th, 7 AM PST
  • How to QUICKLY Solve GMAT Questions - GMAT Club Chat

     November 20, 2018

     November 20, 2018

     09:00 AM PST

     10:00 AM PST

    The reward for signing up with the registration form and attending the chat is: 6 free examPAL quizzes to practice your new skills after the chat.

The random variable x has the following continuous probability distrib

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50621
The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 08 Jul 2018, 04:27
10
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

15% (01:35) correct 85% (01:53) wrong based on 153 sessions

HideShow timer Statistics

The random variable x has the following continuous probability distribution in the range \(0 ≤ x ≤\sqrt{2}\), as shown in the coordinate plane with x on the horizontal axis:

Image

The probability that x < 0 = the probability that \(x > \sqrt{2} = 0\).

What is the median of x?


A. \(\frac{\sqrt{2} - 1}{2}\)

B. \(\frac{\sqrt{2}}{4}\)

C. \(\sqrt{2} - 1\)

D. \(\frac{\sqrt{2} + 1}{4}\)

E. \(\frac{\sqrt{2}}{2}\)


Attachment:
BK8gXRh.jpg
BK8gXRh.jpg [ 13.65 KiB | Viewed 1477 times ]

_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7035
Re: The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 08 Jul 2018, 05:03
1
Bunuel wrote:
The random variable x has the following continuous probability distribution in the range \(0 ≤ x ≤\sqrt{2}\), as shown in the coordinate plane with x on the horizontal axis:

Image

The probability that x < 0 = the probability that \(x > \sqrt{2} = 0\).

What is the median of x?


A. \(\frac{\sqrt{2} - 1}{2}\)

B. \(\frac{\sqrt{2}}{4}\)

C. \(\sqrt{2} - 1\)

D. \(\frac{\sqrt{2} + 1}{4}\)

E. \(\frac{\sqrt{2}}{2}\)


Attachment:
The attachment BK8gXRh.jpg is no longer available


An interesting question...

The frequency of x is given by the slanting line and x is between 0 and √2

Now the median will require knowing the centre point of this so formed triangle with x axis and y axis..

It will happen where a line drawn horizontally divides the so formed triangle in two equal areas..

Area of bigger triangle = 1/2 *√2*√2=1
So the are of smaller triangle should be 1/2
And let this happen at distance A from √2. Since the triangles - smaller and bigger - both are similar, the vertical line will also be A.

Area now =1/2 *A*A=1/2....A^2=1...A=1
The x axis is √2 and median lies 1 away from √2, so √2-1

C
Attachments

PicsArt_07-08-06.25.35.jpg
PicsArt_07-08-06.25.35.jpg [ 9.12 KiB | Viewed 1364 times ]


_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

MBA Section Director
User avatar
D
Affiliations: GMATClub
Joined: 22 May 2017
Posts: 994
Concentration: Nonprofit
GPA: 4
WE: Engineering (Computer Software)
The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 08 Jul 2018, 05:15
Bunuel wrote:
The random variable x has the following continuous probability distribution in the range \(0 ≤ x ≤\sqrt{2}\), as shown in the coordinate plane with x on the horizontal axis:

The probability that x < 0 = the probability that \(x > \sqrt{2} = 0\).



Bunuel,

The question statement says \(0 ≤ x ≤\sqrt{2}\) and from the graph, the probability of x being \(\sqrt{2}\) is 0. So I guess the question statement should say \(0 ≤ x <\sqrt{2}\) ?
_________________

If you like my post press kudos +1

New project wSTAT(which Schools To Apply To?)

New - RC Butler - 2 RC's everyday

Director
Director
User avatar
P
Status: Learning stage
Joined: 01 Oct 2017
Posts: 930
WE: Supply Chain Management (Energy and Utilities)
Premium Member
Re: The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 08 Jul 2018, 05:22
The probability that x < 0 = the probability that \(x > \sqrt{2} = 0\).

Hi chetan2u Sir,
Could you please explain the significance of the highlighted information provided in the question?

And how this info is helpful in solving the question?

Thanking you.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7035
Re: The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 08 Jul 2018, 05:32
PKN wrote:
The probability that x < 0 = the probability that \(x > \sqrt{2} = 0\).

Hi chetan2u Sir,
Could you please explain the significance of the highlighted information provided in the question?

And how this info is helpful in solving the question?

Thanking you.


Same wine in different bottle
The point tells us that the left edge of x is 0 and right edge is √2.
Although same info is also stated by 0<=x<=√2, but we should be able to make out the redundant matter and not pay attention on it.
If someone starts spending time between the two statements pointing towards same thing, the Question / test makers aim is achieved.

So read the info as gi en and discard what is not important or what gives you the same wine in a different bottle.
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

GMATH Teacher
User avatar
G
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 477
Re: The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 27 Sep 2018, 10:53
1
Bunuel wrote:
The random variable x has the following continuous probability distribution in the range \(0 ≤ x ≤\sqrt{2}\), as shown in the coordinate plane with x on the horizontal axis:

Image

If Prob(\(x < 0\)) = Prob(\(x > \sqrt{2}\)) = 0, what is the median of x?

A. \(\frac{\sqrt{2} - 1}{2}\) \(\,\,\,\,\,\,\,\) B. \(\frac{\sqrt{2}}{4}\) \(\,\,\,\,\,\,\,\) C. \(\sqrt{2} - 1\) \(\,\,\,\,\,\,\,\) D. \(\frac{\sqrt{2} + 1}{4}\) \(\,\,\,\,\,\,\,\) E. \(\frac{\sqrt{2}}{2}\)

We are talking about the median of a continuous random variable with a given probability density function... and all this wording is absolutely out-of-GMAT´s-scope!

On the other hand, the same problem could be put into GMAT´s reality (for instance) like that:

Quote:
In the figure above, what is the value of x such that the areas under the line from 0 to x, and from x to \(\sqrt{2}\), coincide?


Image

\(? = x\,\,\,\,:\,\,\,\,{S_{\,{\text{I}}}} + {S_{\,{\text{II}}}} = {S_{\,{\text{III}}}}\,\,\,\,\,\,\left( * \right)\)

\(\left( * \right)\,\,\,\, \Rightarrow \,\,\,\,{S_{\,{\text{I}}}} + {S_{\,{\text{II}}}} = \frac{{{S_{\,{\text{total}}}}}}{2} = \frac{1}{2}\left( {\frac{{\sqrt 2 \cdot \sqrt 2 }}{2}} \right) = \frac{1}{2}\)

\(\frac{1}{2} = {S_{\,{\text{I}}}} + {S_{\,{\text{II}}}} = \frac{{x \cdot x}}{2} + x\left( {\sqrt 2 - 1} \right) = x\sqrt 2 - \frac{{{x^2}}}{2}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,2x\sqrt 2 - {x^2} = 1\,\)


Now we have at least two possible continuations:


First: test alternatives (easy but not so quick).


Second: solve the second-degree equation (need "insight", but VERY quick), as follows (for instance):

\(2x\sqrt 2 - {x^2} = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^2} - 2x\sqrt 2 + 1 = 0\)

\({\text{Sum}}\,\, = \,\,\,2\sqrt 2 \,\,\,\,\,,\,\,\,\,{\text{Product}}\,\, = \,\,\,1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x_1} = \sqrt 2 - 1\,\,,\,\,\,{x_2} = \sqrt 2 + 1\)

\({x_2} > \sqrt 2 \,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {x_1} = \sqrt 2 - 1\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

GMATH Teacher
User avatar
G
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 477
Re: The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 27 Sep 2018, 11:19
workout wrote:
Bunuel,
The question statement says \(0 ≤ x ≤\sqrt{2}\) and from the graph, the probability of x being \(\sqrt{2}\) is 0. So I guess the question statement should say \(0 ≤ x <\sqrt{2}\) ?


Hi, workout!

I have taken the liberty of answering your question. I hope you both (Bunuel and yourself) don´t mind!

When dealing with continuous random variables, the probability of choosing ANY particular value is ALWAYS zero, therefore it really doesn´t matter if you consider to include, or not, any of the interval extremities...

When dealing with continuous random variables, it is useful to consider intervals, not "points", for example(s):

What is the probability that you will buy a light bulb that will not burn-out in less than 2h?

In this case, you are interested in the interval [0,2] , or ]0,2] , or [0,2[ , or ]0,2[ ...

Curiosity: in this situation, it is commonly considered an exponential distribution (see image below).

Image

(The graphs are related to the density and accumulated functions.)

Regards,
Fabio.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Intern
Intern
avatar
B
Joined: 04 Apr 2017
Posts: 19
Re: The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 18 Oct 2018, 13:19
Hi fskilnik

I've a problem with the usage of the term "median" here. For e.g. the median of set A (1,2,100) is 2 although 100 not equal 1. So in our case, the median of x must be the value corresponding to x= root2/2. What's wrong here?
GMATH Teacher
User avatar
G
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 477
The random variable x has the following continuous probability distrib  [#permalink]

Show Tags

New post 18 Oct 2018, 16:00
1
HisHo wrote:
Hi fskilnik

I've a problem with the usage of the term "median" here. For e.g. the median of set A (1,2,100) is 2 although 100 not equal 1. So in our case, the median of x must be the value corresponding to x= root2/2. What's wrong here?

Hi, HisHo.

Thank you for your interest in the subject, even with my explicit mention that it is out of GMAT´s scope.

(I hope I understood your question. If not, please ask again, no problem at all.)

It is NOT as if x denotes the "position" in a list and y denotes the corresponding value at that position.

In this context, the vertical axis offers probabilities associated with values (intervals of values) in the horizontal ("x") axis.

When we are asked the median value of x, it is really the median value of the possible values of x, taking into account the "frequency" of these values that is given (shown) in the vertical axis. (This sentence will be explained more explicitly below.)

If you understood my last sentence, now it must be clear the answer is not:

(1) y=f(a) , where a is (square root of 2)/2
(2) y=f(b) , for any other b

In both cases because y does not have this "value" property associated with the position that would be given in the x axis.
(This is not what is going on, so to speak!)

(3) (square root of 2)/2 , because the "distribution of frequencies" is "irregular", I mean, the smaller (positive) values of x are more frequent...
Consequence: the value x=m (the median) that divides the "list" of x´s (the whole interval from 0 to the square root of 2) such that,
the probability of randomly choosing a number in the interval [0,m] is 50% , is less the (square root of 2)/2 ... got it?

I hope you understood (and liked) my INFORMAL comments.

Regards and success in your studies,
Fabio.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

GMAT Club Bot
The random variable x has the following continuous probability distrib &nbs [#permalink] 18 Oct 2018, 16:00
Display posts from previous: Sort by

The random variable x has the following continuous probability distrib

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.