# Thread: Likelihood function derivation help needed

1. ## Likelihood function derivation help needed

struggling with this. can anyone point me in the right direction?

see attached, i couldnt write down the function at the bottom in here to made it and had to attach it.

2. Originally Posted by da`
struggling with this. can anyone point me in the right direction?

see attached, i couldnt write down the function at the bottom in here to made it and had to attach it.
$L \propto \left[ exp\left(-\frac{x_1^2}{2}\right) \cdot exp\left(-\frac{x_2^2}{2}\right) \cdot \, .... \cdot exp\left(-\frac{x_{\theta}^2}{2}\right) \right]$ $\cdot \left[exp\left(-\frac{(x_{\theta + 1}^2 - 1)}{2}\right) \cdot \, .... \cdot exp\left(-\frac{(x_{n}^2 - 1)}{2}\right) \right]$

$= exp \left(- \sum_{i=1}^{\theta} \frac{x_i^2}{2} \right) \cdot exp \left(- \sum_{i=\theta + 1}^{n} \frac{(x_i^2 - 1)}{2} \right)$

$= exp \left(-\sum_{i=1}^{\theta} \left[\frac{(x_i - 1)^2}{2} + \frac{(2x_i - 1)}{2}\right] \right) \cdot exp \left(- \sum_{i=\theta + 1}^{n} \frac{(x_i^2 - 1)}{2} \right)$

$= exp \left(- \sum_{i=1}^{n} \frac{(x_i - 1)^2}{2}\right) \cdot exp \left(- \sum_{i=1}^{\theta} \frac{(2x_i - 1)}{2} \right)$

$\propto exp \left(- \sum_{i=1}^{\theta} \frac{(2x_i - 1)}{2} \right)$.

3. Thank you very much, i understand.