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Let $a,b,c, \tau$ be positive constants and $x$ is an exponentially distributed variable with parameter $\lambda = 1$, i.e. $x\sim\exp(1)$.

$$E = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+$$

where $[z]^+ = \max(z,0)$

How can I find

  1. The PDF for $E$
  2. The expectation of $E$.
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first calculate the interval where $Z=\frac{a(a+1)}{1+e^{-bX+c}}-1\leq0$

The probability of this interval is a mass probability assigned to $Z=0$. The remainig probability is distributed over $Z$ according to the specified random law. The resulting "PDF" is a mixed density, not absolutely continuous.

Finally calculate the density of $Y=\tau[Z]^+$ by a monotonic transformation and its expectation in the way you are used to do

Do not forget that Random Variables need Capital Letters...

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