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ChrIsPuZa

Member

Registered: 2016-05-03

Posts: 8

a probability question.

suppose random variables X and Y has joint pdf  f(x,y)=k-y   0<y<x^2<1

a)  find k

b) find marginal pdf f(x)  and the conditional pdf f(x|1/2)  and cdf F(y|1/2);

c) find P(X<Y|Y>1/2), E(X|1/2)  and E(X|Y>1/2)

d) what's the correlation between X and Y?

e) find the value of pdf of U=X+Y evaluated at u=0.8 and estimate P(0.8<X+Y<0.801)

Any help here would be greatly appreciated!

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: a probability question.

suggests the region below the curve

,ordinates x=0 and x=1 and y=0 the x axis.
If the pdf is integrated over this region it should give a value of 1. therefore,

there are some errors in this. see later comments.

Last edited by thickhead (2016-05-15 00:58:09)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

ChrIsPuZa

Member

Registered: 2016-05-03

Posts: 8

Re: a probability question.

Hi thickhead:

fantastic  answer. many thanks.

I still have no idea about (e),how to estimate P(0.8<X+Y<0.801)?

jakndaxta

Member

Registered: 2016-03-15

Posts: 18

Re: a probability question.

Hi thickhead,

I'm having trouble starting to calculate the cdf F(y|1/2), P(X<Y|Y>1/2), and E(X|Y>1/2).

Just wondering if you could show me how to do those after getting back to ChrIsPuZa?

Thanks!

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: a probability question.

I am bit perplexed.are X and Y synonymous with x an y?
Actually I have no expertise in the topic . I know elementary probability but not the advanced one. I found the definition of conditional probability on the net and solved that. what about cdf F(y|1/2) Is it conditional probability or cumulative probability? x<y is outside the region stipulated and hence probability seems to be undefined. i am not sure.

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

jakndaxta

Member

Registered: 2016-03-15

Posts: 18

Re: a probability question.

Hi thickhead,

F(y|1/2) is a conditional cdf of y on y=1/2 (at least I'm assuming the second part is y). Also for part a would we have to account for the case where x is negative?

Thanks!

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: a probability question.

suggests the region below the curve

,ordinates x=0 and x=1 and y=0 the x axis.
If the pdf is integrated over this region it should give a value of 1. therefore,

I committed  some errors in what I wrote above. because it is 0<x^2<1 the region encompasses the part below parabola in 2nd quadrant also.

suggests the region below the curve

,ordinates x=-1 and x=1 and y=0 the x axis.
If the pdf is integrated over this region it should give a value of 1. therefore,

Last edited by thickhead (2016-05-15 01:05:35)

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

jakndaxta

Member

Registered: 2016-03-15

Posts: 18

Re: a probability question.

Okay thanks, that was confusing me a bit!

Did you have any thoughts on the other questions?

EDIT: I get k=9/5(=1.8) when solving the first equation

Last edited by jakndaxta (2016-05-14 19:22:56)

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: a probability question.

You are right k=1.8 . i had merely halved what I had got earlier.

Now for P(x<Y|Y>1/2) the area below the parabola in 2nd quadrant qualifies for X<Y. It can be calculated as

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: a probability question.

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

jakndaxta

Member

Registered: 2016-03-15

Posts: 18

Re: a probability question.

Thank you for your help, thickhead!