Results 1 to 2 of 2

Math Help - urgent hw....do help....probability densities

  1. #1
    souhar
    Guest

    urgent hw....do help....probability densities

    Hi i have the following problems as HW.......try to help me......urgent.......

    1) If X and Y have a bivariate normal distribution and U=X+Y and V=X-Y find an expression for the correlation coefficient of U and V.
    2)If X has an exponential distribution,show that
    P(X >= t + T \ X >= T) = P(X >= t)
    this property of an exponential random variable parallels that of a geometric random variable as [P(X = x+n \ X > n)] = P (X=x)
    3)If rando variable T is the time to failure of a commercial product and the values of its probability dnsity and distribution function at time "t" are f(t) and F(t), then its failure rate at time t is given by f(t) / 1-F(t). Thus the failure rate at time t is the probability density of failure at time t given that failure does not occur prior to time t.
    a) show that if T has an exponential distribution, the failure rate is constant.
    b)show that if has a weibull distribution, the failure rate is given by ab t^b-1.

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by souhar View Post
    1) If X and Y have a bivariate normal distribution and U=X+Y and V=X-Y find an expression for the correlation coefficient of U and V.
    <br />
\rho_{U,V} = \frac{E((U-\bar{u})(V-\bar{v}))}{\sigma_U\sigma_V}=<br />
<br />
\frac{E((X+Y-\bar{x}+\bar{y})(X-Y-\bar{x}+\bar{y}))}{\sigma_U\sigma_V}=<br />
<br />
\frac{E((X-\bar{x})^2+(Y-\bar{y})^2)}{\sigma_U\sigma_V}=<br />
<br />
\frac{\sigma_X^2+\sigma_Y^2}{\sigma_U\sigma_V}<br />

    Now:

    \sigma^2_U = E((U-\bar{u})^2)= <br />
E( [(X+Y) - (\bar{x}+\bar{y})]^2)= <br />
E((X-\bar{x})^2 + 2(X-\bar{x})(Y-\bar{y}) +(Y-\bar{y})^2)= <br />
\sigma_X^2 + 2\sigma_{X,Y} + \sigma_Y^2<br />

    A similar argument gives:

    \sigma^2_U =<br />
\sigma_X^2 - 2\sigma_{X,Y} + \sigma_Y^2<br />

    and I leave the rest to you.

    (By the way I have not used the information that X and Y are bivariate normal)

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Convergence of densities in Lindeberg's CLT
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 18th 2011, 06:49 PM
  2. Replies: 6
    Last Post: June 9th 2010, 01:01 AM
  3. Joint Densities
    Posted in the Statistics Forum
    Replies: 1
    Last Post: April 1st 2010, 02:58 AM
  4. probability: conditioning and densities
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: December 4th 2007, 05:01 AM
  5. joint densities
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 14th 2007, 03:20 AM

Search Tags


/mathhelpforum @mathhelpforum