You are Here: Home >< Maths

# Obtaining a PDF and CDF of random variable Y watch

1. 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/Turbulen...apter3/3.9.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.
2. (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,

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.
3. (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,

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)?

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: March 11, 2016
The home of Results and Clearing

### 1,441

people online now

### 1,567,000

students helped last year
Today on TSR

### University open days

1. Bournemouth University
Wed, 22 Aug '18
2. University of Buckingham
Thu, 23 Aug '18
3. University of Glasgow
Tue, 28 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams