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Obtaining a PDF and CDF of random variable Y

I have a random variable, U with a known PDF, f(V) and a known CDF, F(V).

I have another random variable, Y, defined as:

Y=Q(U).

Q is a function and is either:
a) Monotonic increasing.
b) Monotonic decreased.

So how do I obtain the PDF and CDF of Y?

The entire question is:



Eq. 3.50 and 3.51 (Y defined below is log-normal, but it is not necessarily log-normal in the question):


Well the question advises on how to start the question and the solution is given here:
http://pope.mae.cornell.edu/TurbulentFlows/solutions/solutions/chapter3/3.9.pdf

What I don't understand is why in line 6, the sign of the probability changes. I cannot picture how the function, Q, or its inverse, is changing the sample space, y. Can anyone enlighten me please?

Also, just comparing the answers, Eq. 3.61 shows the CDF becomes symmetric about the x axis, so wouldn't this create a monotonic decreasing CDF which is prohibited because of how the CDF is defined?

Thanks.
(edited 8 years ago)
Original post by djpailo

What I don't understand is why in line 6, the sign of the probability changes. I cannot picture how the function, Q, or its inverse, is changing the sample space, y. Can anyone enlighten me please?

Also, just comparing the answers, Eq. 3.61 shows the CDF becomes symmetric about the x axis, so wouldn't this create a monotonic decreasing CDF which is prohibited because of how the CDF is defined?
Thanks.


I must admit that I find the notation used in this question truly horrible. I'm not surprised that you are confused.

The answer to both of your question is that the function Q is monotonically decreasing. So for the first,

P(Q(x)<y)=P(X>Q−1(y) \displaystyle \mathbb{P}(Q(x) < y) = \mathbb{P}(X > Q^{-1}(y)

Draw a picture of a montone decreasing function and look at where these respective conditions are true. For the second, notice that as the function is decreasing, increasing the x coordinate decreases the y coordinate and hence the cumulative function runs backwards, as it were.
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Original post by Gregorius
I must admit that I find the notation used in this question truly horrible. I'm not surprised that you are confused.

The answer to both of your question is that the function Q is monotonically decreasing. So for the first,

P(Q(x)<y)=P(X>Q−1(y) \displaystyle \mathbb{P}(Q(x) < y) = \mathbb{P}(X > Q^{-1}(y)

Draw a picture of a montone decreasing function and look at where these respective conditions are true. For the second, notice that as the function is decreasing, increasing the x coordinate decreases the y coordinate and hence the cumulative function runs backwards, as it were.


So drawing say Q(V) against V and assuming Q is a linear function (of the form y=-mx + c) and decreasing, would it be correct in saying that for increasing values of the sample space, V, we get smaller values for the sample space, Q(V), where Q(V) = y is the sample space for the random variable, Y.

Then if that is the case, do I picture the CDF as still, a monotonic increasing function, but as the sample space for the random variable, Y, goes from left to right, it goes from positive infinity to negative infinity (as opposed to normally being negative infinity to positive infinity going left to right)?
(edited 8 years ago)

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