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Thread: Random variables - density functions

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    Question Random variables - density functions

    (a) If Z1 and Z2 are independent and have N(0,1) densities, let

    X1 = a11Z1 + a12Z2 + c1 and let X2 = a21Z1 + a22Z2 + c2

    where a11,a12,a21,a22,c1 and c2 are constants. Determine the joint density of X1 and X2.


    (b) Suppose that the random variable X on (0,1) has density

    f(x) = 3x2 0 < x < 1

    Determine the density functions of

    (i) U = X1/2
    (ii) V = -log(X)

    (c) Now suppose Y is uniformly distributed on (0,1) and independent of X. Determine the density of

    W = max(X,Y)
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  2. #2
    Member Last_Singularity's Avatar
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    Quote Originally Posted by GenericNameA View Post
    (a) If Z1 and Z2 are independent and have N(0,1) densities, let

    X1 = a11Z1 + a12Z2 + c1 and let X2 = a21Z1 + a22Z2 + c2

    where a11,a12,a21,a22,c1 and c2 are constants. Determine the joint density of X1 and X2.


    (b) Suppose that the random variable X on (0,1) has density

    f(x) = 3x2 0 < x < 1

    Determine the density functions of

    (i) U = X1/2
    (ii) V = -log(X)

    (c) Now suppose Y is uniformly distributed on (0,1) and independent of X. Determine the density of

    W = max(X,Y)
    Part B (i):
    We are given that $\displaystyle f_X(x) = 3X^2, X \in [0,1]$. And given that $\displaystyle U = X^{1/2}$, we need the distribution of $\displaystyle U$

    Step 1: Determine the domain of the random variable $\displaystyle U$. Since $\displaystyle X$ is at most $\displaystyle 1$, $\displaystyle U$ is at most $\displaystyle 1$. And similar, because $\displaystyle X$ is at least $\displaystyle 0$, $\displaystyle U$ is at least $\displaystyle 0$.
    So $\displaystyle U \in [0,1]$ as well.

    Step 2: find the CDF of $\displaystyle U$: $\displaystyle F_U(u)$
    $\displaystyle = P(U < u)$

    $\displaystyle = P(X^{1/2} < u)$

    $\displaystyle =P(X < u^2)$

    $\displaystyle =F_X(u^2)$

    Step 3: find the pdf of $\displaystyle U$
    And since $\displaystyle f_U(u) = \frac{\partial F_U}{\partial U}$ and we concluded that $\displaystyle F_U(u) = F_X(u^2)$, it must be the case that:

    $\displaystyle f_U(u) = \frac{\partial F_X}{\partial U}$

    $\displaystyle = f_x(u^2) \frac{\partial (u^2)}{\partial u}$

    $\displaystyle =2u f_x(u^2)$

    $\displaystyle =2u 3 (u^2)^2$

    $\displaystyle =6u^5$

    Step 4: Vary that indeed this is a pdf:
    $\displaystyle \int_0^1 6u^5 du = 1$ as desired.

    Part B (ii):
    This one is a bit tougher. But we can proceed as before:

    Step 1: The boundaries of $\displaystyle X$ are $\displaystyle 0,1$. So if we let $\displaystyle V = - log(x)$, then the boundaries of $\displaystyle V$ become $\displaystyle 0$ and positive infinity.
    So $\displaystyle V \in [0, \infty]$

    Step 2: find the CDF of $\displaystyle V$: $\displaystyle F_V(v)$
    $\displaystyle = P(V < v)$

    $\displaystyle = P(-log(X) < v)$

    $\displaystyle = P(10^{-log(X)} < 10^v)$

    $\displaystyle = P(1/X < 10^v)$

    $\displaystyle = P(X < 10^{-v})$

    $\displaystyle = F_X(10^{-v})$

    Step 3: find the PDF of $\displaystyle V$: $\displaystyle f_V(v)$
    $\displaystyle = \frac{\partial F_X(10^{-v})}{\partial v}$

    $\displaystyle = -ln(10)(10^{-v}) f_x(10^{-v})$

    $\displaystyle =3 ln(10) (1/100)^v (10^{-v})$

    Step 4: verify that this is indeed a valid pdf:
    $\displaystyle \int_0^\infty 3 ln(10) (1/100)^v (10^{-v}) dv = 1$ as desired.

    Part C:
    We proceed as before by finding the cdf of $\displaystyle W$:
    $\displaystyle = P(W < w)$

    Now, here's the trick. Since we have defined $\displaystyle W = max(X,Y)$ or that $\displaystyle W$ represents the larger of $\displaystyle X$ and $\displaystyle Y$, it means that whenever $\displaystyle W$ is less than a constant $\displaystyle w$, then it must be the case that both $\displaystyle X$ and $\displaystyle Y$ are less than that constant $\displaystyle w$. So we can determine that:
    $\displaystyle = P(W < w)$
    $\displaystyle = P(X < w, Y < w)$
    $\displaystyle = P(X < w) P(Y < w)$ (by independent of $\displaystyle X$ and $\displaystyle Y$)
    $\displaystyle = F_X(w) F_Y(w)$

    We know that $\displaystyle F_X(x) = \int f_X(x) dx = \int 3X^2 dx= X^3$ and that $\displaystyle F_Y(y) = \int f_Y(y) dy = \int 1 dy = Y$

    So $\displaystyle F_X(w) F_Y(w) = (W^3)(W) = W^4$

    We conclude that the cdf of $\displaystyle W$ is: $\displaystyle F_W(w) = W^4$

    Then, of course, the pdf of $\displaystyle W$ is: $\displaystyle f_W(w) = 4W^3, W \in [0,1]$

    Verify that this is a valid pdf and you are done.

    Part A:
    I am too tired lol...just scroll to the section that says "bivariate case" at: http://en.wikipedia.org/wiki/Bivariate_normal

    Plug in the appropriate variables into the formula...
    Last edited by Last_Singularity; Jan 14th 2009 at 07:25 PM.
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