Question-19

MLE

Let $X$ be a random variable with PDF: \[ $f_X(x) = (\lambda a)x^{\alpha -1}e^{-\lambda x^{\alpha}}, \hspace{5mm} x>0,$ \] where $\alpha$ and $a$ are constants. Find the maximum likelihood estimator of $\lambda$ for $n$ i.i.d. samples of $X$.

$\dfrac{\sum\limits_{i=1}^{n}X_i^{\alpha}}{n}$
$\dfrac{n}{\sum\limits_{i=1}^{n}X_i^{\alpha}}$
$\dfrac{n}{\alpha \sum\limits_{i=1}^{n}X_i^{\alpha}}$
$\dfrac{\sum\limits_{i=1}^{n}X_i^{\alpha}}{n\alpha }$

Hint

Use the concept of MLE (Maximum Likelihood Estimator)

Answer

$\dfrac{\sum\limits_{i=1}^{n}X_i^{\alpha}}{n}$
$\dfrac{n}{\sum\limits_{i=1}^{n}X_i^{\alpha}}$
$\dfrac{n}{\alpha \sum\limits_{i=1}^{n}X_i^{\alpha}}$
$\dfrac{\sum\limits_{i=1}^{n}X_i^{\alpha}}{n\alpha }$

Solution