# Thread: Derivation in paper - calculus - How?

1. ## Derivation in paper - calculus - How?

Hi all

the paper is Pybus(2000) An integrated framework for the inference of viral population history from reconstructed genealogies

wanted some help with this

$\displaystyle U = exp(-\int^{gi+ti}_{x=ti} \frac{{i \choose 2}}{Ne(x)} dx$

In the paper they solve U for gi and get

$\displaystyle g_i {i \choose 2} = -ln(U) \cdot ~ (\int^{gi+ti}_{x=ti} \frac{1}{Ne(x)}\frac{dx}{gi})^{-1}$

i dont know how he has done the re arranging. Im not even sure it is correct

can someone please show me how? or if its too much effort tell me what method he used

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for those interested U is a uniform unit random variable. Solving for gi will generate a variate sample from an equation.

Thank you guys so much for the help

2. The exponent on the integral expresssion gives it away.

After the logarithm is introduced, simply multiply and divide the integral side by $\displaystyle g_{i}$. Leave the numerator $\displaystyle g_{i}$ outside the integral expression and put the denominator $\displaystyle g_{i}$ inside the integral expression.

$\displaystyle i \choose 2$ is also constant and has been removed from the integral expression before division of the integral expression to the other side.

Watch out for these things. In practical applications, integrals, derivative, logarithms, and various other things sometimes are substituted with some form of linear approximation. That does not appear to have happened here, but keep your eyes out for it in your future readings.