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Strange Stats Question

U is a continuous random variable on the range (0,1). A random variable X is defined by X=1/(1+6.5U).

Calculate the lower quartile of X and the expectation of X and add them together, reporting your result to 3d.p

Now I have no idea what to do for this one. I though of integrating between 0,1 of 1/(1+6.5U)

Please advice or give clue

Thanks
Reply 1
Original post by mathsRus
U is a continuous random variable on the range (0,1). A random variable X is defined by X=1/(1+6.5U).

Calculate the lower quartile of X and the expectation of X and add them together, reporting your result to 3d.p

Now I have no idea what to do for this one. I though of integrating between 0,1 of 1/(1+6.5U)

Please advice or give clue

Thanks


if U is uniformly distributed in (0,1) the X is also uniformly distributed in (1/(1+6.5),1/(1+0))

i.e in (1/7.5,1)
Reply 2
so do I integrate in terms of X but between (1/7.5,1)?
Reply 3
Original post by mathsRus
so do I integrate in terms of X but between (1/7.5,1)?


So do I integrate w.r.t.x = so it becomes U = 1/6.5X - 1/6.5

and integrate between range you specified.

Also when it comes to lower quartile how would I solve q1 - ln(q1) = 1.625
Original post by TeeEm
if U is uniformly distributed in (0,1) the X is also uniformly distributed in (1/(1+6.5),1/(1+0))

i.e in (1/7.5,1)


With all due respect, X is definitely not uniformly distributed.
(edited 9 years ago)
Reply 5
Original post by ghostwalker
With all due respect, X is definitely not uniformily distributed.


I thought about it after I typed (one glass of red wine sipped)
now after almost three... feel free to correct and have no respect
Original post by TeeEm
I thought about it after I typed (one glass of red wine sipped)
now after almost three... feel free to correct and have no respect


Didn't really want to get involved in this thread, as I'd have to work things out from scratch, and don't have the time, but couldn't let your statement stand.
Reply 7
I thought it was uniform for once but how would I go about doing it now?
Reply 8
Original post by mathsRus
I thought it was uniform for once but how would I go about doing it now?


I would not like to help you because I am not in a fit state at present, but I think Ghostwalker knows stats and he has to be right.

In the back of my head I can only think of working this out from the CDFs

it goes a bit like this

let F(u) be the CDF of the uniform
F(u) = u

let G(x) be the CDF of X

G(x)=P(X<x) = P[1/(1+6.5U)<x) = P(U>[(1-x)/(6.5x)] = 1 - F[(1-x)/(6.5x)] = 1- (1-x)/(6.5x) = ... = 15/13 - 2/(13x)


differentiate and g(x) = 2/(13x^2) limits 2/15 to 1




111.jpg
Original post by mathsRus
...


OK, my brain is working again this morning.

X is a function of U, so E(X) is simply 01x(u)fu  du\displaystyle\int_0^1x(u)f_u\;du
where x(u) is the function you've been given and f_u is the pdf of U, which is simply "1" here.


For the lower quartile, since your function is monotonically decreasing, the lower quartile of X is going to correspond with the upper quartile of U. I.e., the value of U that gives us the upper quartile for U, will correspond to the value of X that gives us the lower quartile for X.

I.e. qx1=x(qu3)q_{x1}=x(q_{u3})

Hope that made sense - once you see it, it's clear, but I'm finding it difficult to explain.
Original post by TeeEm

but I think Ghostwalker knows stats


Up to a point - there are plenty of more knowledgeable people on here.


and he has to be right.


If only I could get the rest of the world to agree with you - life would be so much easier. :smile:
Reply 11
Original post by mathsRus
U is a continuous random variable on the range (0,1). A random variable X is defined by X=1/(1+6.5U).

Calculate the lower quartile of X and the expectation of X and add them together, reporting your result to 3d.p

Now I have no idea what to do for this one. I though of integrating between 0,1 of 1/(1+6.5U)

Please advice or give clue

Thanks


I did recheck my method shown in post 9 and I feel it is correct

once you have the pdf of X you can work out everything including Expectation, quartiles etc

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