Turn on thread page Beta

Obtaining a PDF and CDF of random variable Y watch

    • Thread Starter
    Offline

    20
    ReputationRep:
    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.
    Online

    14
    ReputationRep:
    (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,

     \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.
    • Thread Starter
    Offline

    20
    ReputationRep:
    (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,

     \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)?
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: March 11, 2016
The home of Results and Clearing

1,441

people online now

1,567,000

students helped last year

University open days

  1. Bournemouth University
    Clearing Open Day Undergraduate
    Wed, 22 Aug '18
  2. University of Buckingham
    Postgraduate Open Evening Postgraduate
    Thu, 23 Aug '18
  3. University of Glasgow
    All Subjects Undergraduate
    Tue, 28 Aug '18
Poll
How are you feeling about GCSE results day?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.