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Thread: P(Z<z), Z=X if X<=1, Z=Y+1 if X>1

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    P(Z<z), Z=X if X<=1, Z=Y+1 if X>1




    So here is the question

    X∼exp(1)X
    Y∼exp(2)Y (exponentialy distributed)
    X and y are independent.

    Z=Xif x≤1. Otherwise Z=Y+1. What is the cumulative distribution function of Z?

    So what i thought.
    Fz(z)=P(Z<z)=P(Z<z∣X<1)P(X<1)+P(Z<z∣X>1)P(X>1)=(1−e−z)(1−e−1)+(1−e−2(z−1))

    But i am realy not sure. Should z maybe be devided in to tow cases wher z<1 and z>1 maybe? So F(z) is not continouse?
    I am lost. Please help.
    Last edited by ogry; 01-16-2016 at 01:53 AM.

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    Re: P(Z<z), Z=X if X<=1, Z=Y+1 if X>1

    can anyone tell me if this is possible? please I've been breaking my head over this for 2 days. and I am actualy pretty sure its not suppose to be that hard.

    F(z) = (1-e^-z)(1-e^-1) if Z<=1
    F(z)=(1-e^-2(z-1))e^-1 if z>1

    or
    maybe it would be
    F(z)=(1-e^-z)(1-e^-1)+(1-e^-2(z-1))e^-1
    both of them I derive from the total probability rule but one is divided into 2 parts and on is not.
    but both don't feel right and I have no idea how else to approach it.
    Last edited by ogry; 01-17-2016 at 06:07 AM.

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    Re: P(Z<z), Z=X if X<=1, Z=Y+1 if X>1


    The given question is X \sim \text{Exp}(1), Y \sim \text{Exp}(2), X, Y are independent. Assuming the parametrization means the pdf of Y is f_Y(y) = 2e^{-2y}, y > 0. Define

    Z = \begin{cases} X &\text{if } X \leq 1 \\
Y + 1 & \text{if } X > 1 \end{cases}

    It is trivial that \Pr\{Z \leq 0\} = 0. By law of total probability, for z > 0, the CDF F_Z(z)

    = \Pr\{Z \leq z\}

    = \Pr\{Z \leq z, X \leq 1\} + \Pr\{Z \leq z, X > 1\}

    = \Pr\{X \leq z, X \leq 1\} + \Pr\{Y + 1 \leq z, X > 1\}

    = \Pr\{X \leq \min\{z, 1\}\} + \Pr\{Y \leq \max\{z - 1, 0\}\}\Pr\{X > 1\}

    = 1 - e^{-\min\{z, 1\}} 
+ \left(1 - e^{-2(z - 1)}\right)e^{-1}\mathbf{1}\{z > 1\}

    = \begin{cases} 1 - e^{-z} & \text{if } 0 < z \leq 1 \\
1 - e^{-1} + e^{-1} - e^{-2z + 1} = 1 - e^{-2z + 1} & \text{if } z > 1
 \end{cases}

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