probability density

[ladka4gals]

Determine the probability density function for the following hazard
rate:
z(t) = a.exp(-bt), a> 0, b >0, t>0. please help me solve this. thanks.

[Quantumcat]

the probability density is the function times its complex conjugate, or at least it is in quantum mechanics :-) Just the square in your case as there are no imaginary parts. |z(t)|^2=a^2exp(-2bt) ?

And if you want to normalise it, take the integral from +inf to -inf make it equal to 1, then a is determined by this :-)

a^2*(Integral (z(t) from 0 to +inf)=[(-1/b)exp(-bt)](0, inf)=1

a^2[(-1/b)- lim(s->inf){exp(-bs)}]=1

a^2(-1/b) = 1

a^2=-b

a = i(sqrt b)

Oops maybe I did something wrong?