Question 393605
The mean {{{1/alpha}}} of the exponential function is 1/2, and hence {{{alpha = 2}}}.  The density function becomes {{{f(x) = alpha*e^(-alpha*x) = 2e^(-2x)}}} for {{{x>=0}}}, and f(x) = 0 otherwise.

{{{P(X < 1) = int( 2e^(-2x), dx, 0, 1) = -e^(-2*1) + e^(-2*0) = 1 - 1/e^2 = 0.86466}}} to 5 decimal places.

{{{P(X < 2.5) = int( 2e^(-2x), dx, 0, 2.5) = -e^(-2*2.5) + e^(-2*0) = 1 - 1/e^5 = 0.99326}}} to 5 decimal places.

{{{P(X > 3) = int( 2e^(-2x), dx, 3, infinity) = 0 + e^(-2*3) = 1/e^6 = 0.002479}}} to 6 decimal places.