Results 1 to 5 of 5

Math Help - Independent Uniform Random Variables

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    179
    Thanks
    1

    Independent Uniform Random Variables

    If X1,X2, . . . ,Xn are independent U(0, 1) random variables and Y=min (X1,X2, . . . ,Xn) has PDF given by

    fY(x) = { n(1-x)^{n-1}, 0<=x<=1}, 0, otherwise

    Identify this distribution, and hence state E[Y] and Var(Y).

    Im thinkin it looks similar to a binomial distribution? Am i on the right lines?
    Last edited by sirellwood; November 24th 2009 at 04:34 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by sirellwood View Post
    If X1,X2, . . . ,Xn are independent U(0, 1) random variables and Y=min (X1,X2, . . . ,Xn) has PDF given by

    fY(x) = { n(1-x)^n-1), 0<=x<=1}, 0, otherwise Mr F says: What you have posted here is wrong. It's not even a pdf ....

    Identify this distribution, and hence state E[Y] and Var(Y).

    Im thinkin it looks similar to a binomial distribution? Am i on the right lines?
    There is a (simple) mistake in your pdf which I'll leave you to figure out and fix.

    The correct pdf has a beta distribution (and you will need to determine what value each parameters has).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2009
    Posts
    179
    Thanks
    1
    Ah thank you mr fantastic! its edited now!

    So i first need to find the sample mean and variance right? Because it is uniformally distributed, is the sample mean just (x1+xn)/2?

    Im getting a little confused with the word "min"(x1,x2....) in the Y term.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    minimum of X1,...,Xn is the smallest of these rvs
    It's also known as the smallest order stat.
    It couldn't be a binomial since the underlying distribution here is continuous (uniform)

    F_Y(a)=P(Y\le a)=1-P(Y>a) for any 0<a<1

    =1- P(X_1>a)P(X_2>a)\cdots P(X_n>a)

    =1- (1-a)^n

    So f_Y(a)={d\over da}F_Y(a)=n(1-a)^{n-1}

    THIS is a Beta with parameters 1 and n.
    Look that up for your mean and variance.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by sirellwood View Post
    Ah thank you mr fantastic! its edited now!

    So i first need to find the sample mean and variance right? Because it is uniformally distributed, is the sample mean just (x1+xn)/2?

    Im getting a little confused with the word "min"(x1,x2....) in the Y term.
    The min business is irrelevant (unless you want to derive the given pdf). The bottom line is that you're given a pdf and asked to identify it and hence get it's mean and variance. Note that you can check your answers by calculating the mean and variance directly from the pdf.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. independent random variables and a uniform distribution...
    Posted in the Advanced Statistics Forum
    Replies: 7
    Last Post: August 5th 2011, 05:13 AM
  2. Sum of Two Independent Random Variables (uniform)
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: February 4th 2011, 01:58 AM
  3. Independent random variables and their PDF
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: September 25th 2010, 08:16 AM
  4. Order Statistics, N independent uniform random variables
    Posted in the Advanced Statistics Forum
    Replies: 9
    Last Post: March 22nd 2009, 10:12 PM
  5. uniform distribution of random independent variables
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: December 1st 2008, 03:45 AM

Search Tags


/mathhelpforum @mathhelpforum