<Dl> <Dd> F Y (y) = P (Y ≤ y) = P (l o g (1 + e − X) ≤ y) = P (X ≥ − l o g (e y − 1)). (\ displaystyle F_ (Y) (y) = P (Y \ leq y) = P (\ mathrm (log) (1 + e ^ (- X)) \ leq y) = P (X \ geq - \ mathrm (log) (e ^ (y) - 1)). \,) </Dd> </Dl> <Dd> F Y (y) = P (Y ≤ y) = P (l o g (1 + e − X) ≤ y) = P (X ≥ − l o g (e y − 1)). (\ displaystyle F_ (Y) (y) = P (Y \ leq y) = P (\ mathrm (log) (1 + e ^ (- X)) \ leq y) = P (X \ geq - \ mathrm (log) (e ^ (y) - 1)). \,) </Dd> <P> The last expression can be calculated in terms of the cumulative distribution of X, (\ displaystyle X,) so </P> <Dl> <Dd> F Y (y) = 1 − F X (− l o g (e y − 1)) (\ displaystyle F_ (Y) (y) = 1 - F_ (X) (- \ mathrm (log) (e ^ (y) - 1)) \,) <Dl> <Dd> <Dl> <Dd> = 1 − 1 (1 + e l o g (e y − 1)) θ (\ displaystyle = 1 - (\ frac (1) ((1 + e ^ (\ mathrm (log) (e ^ (y) - 1))) ^ (\ theta)))) </Dd> <Dd> = 1 − 1 (1 + e y − 1) θ (\ displaystyle = 1 - (\ frac (1) ((1 + e ^ (y) - 1) ^ (\ theta)))) </Dd> <Dd> = 1 − e − y θ . (\ displaystyle = 1 - e ^ (- y \ theta). \,) </Dd> </Dl> </Dd> </Dl> </Dd> </Dl>

How do you find the values of a random variable