plot an exponential statistical function

Question

I have this exponential function, where j is unknown. How can I plot the variance function for $k= 0.1, \; 1, \;10$

$$f_{j,k}(y)=\frac{\sqrt{j}}{\sqrt{2 \pi}}e^{\sqrt{jk}}y^{-\frac{1}{2}} \text{exp}\left( -\frac{1}{2} (j y + \frac{k}{y}) \right) \quad \quad y>0$$

Answer 1

Assuming that given function is a PDF of a distribution:

f[y_, j_, k_] := Sqrt[j]/Sqrt[2 Pi] Exp[Sqrt[j k]]/Sqrt[y] Exp[-(1/2) (j y + k/y)];

You can create an object that will represent your distribution:

distr = ProbabilityDistribution[{"PDF", f[y, j, k]}, {y, 0, Infinity}, 
                                               Assumptions -> (j > 0 && k > 0)];

And now you can calculate anything you want:

Variance[distr]
(*(2 + Sqrt[j k])/j^2*)
Mean[distr]
(*Mean[distr]*)
CentralMoment[distr, 3]
(*(8 + 3 Sqrt[j k])/j^3*)