1. ## 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.

2. Originally Posted by feage7
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.

3. Originally Posted by feage7
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.