# How to transform a Hessian set into the corresponding Monte Carlo PDF

A Hessian PDF set can be transformed into a Monte Carlo replica set using the
method described in Eq. (4.3) of PDF4LHC21 using a runcard
like the one below.
In this example `Neig`

are the number of basis eigenvectors of the Hessian PDF.
`mc_pdf_name`

is the name of the new MC replica set that will be created.
`watt_thorne_rnd_seed`

is the random seed used to generate the MC replicas as shown in the equation below.

where \(f^{MC, (k)}(x,Q)\) is the \(k\)-th Monte Carlo replica, \(f^{H}_{0}(x,Q)\) the central Hessian member, \(f^{H}_{j,\pm}(x,Q)\) the positive and negative eigenvector directions corresponding to the \(j\)-th Hessian eigenvector, and \(R_j^{(k)}\) are random standard normal numbers.

The new MC replica set will be saved in the LHAPDF folder.

```
pdf: MSTW2008nlo68cl
mc_pdf_name: MSTW2008nlo68cl_mc
num_members: 100
watt_thorne_rnd_seed: 1
actions_:
- write_hessian_to_mc_watt_thorne
```