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  1. #1
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    Pdf

    1) The probability density function of a truncated exponential random variable, T, is given by

    f(t) = xe^-xt/e^-xa - e^-xb, x > 0; a<= t <= b

    a) show that this is a valid density function

    b) Given and independant random sample of observations from this truncated exponential distribution, t1,....., tn, say, write down the log-likelihood function for x.
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  2. #2
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    Quote Originally Posted by feage7 View Post
    1) The probability density function of a truncated exponential random variable, T, is given by

    f(t) = xe^-xt/e^-xa - e^-xb, x > 0; a<= t <= b

    a) show that this is a valid density function
    I think I know how to do this question but I am not able to because you are not using paranthesis.
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  3. #3
    Junior Member F.A.P's Avatar
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    Quote Originally Posted by feage7 View Post
    1) The probability density function of a truncated exponential random variable, T, is given by

    f(t) = xe^-xt/e^-xa - e^-xb, x > 0; a<= t <= b

    a) show that this is a valid density function
    If it is a valid density its integral over the real numbers equals 1.

    f(t)=\frac{xe^{-xt}}{(e^{-xa} - e^{-xb})}, x > 0; a\leq t \leq b

    \int_{R}^{}f(t)dt=\int_a^b\frac{xe^{-xt}}{(e^{-xa} - e^{-xb})}dt=-\frac{e^{-xb}-e^{-xa}}{(e^{-xa} - e^{-xb})}=1

    hence f(t) is a valid density.
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