IGNOU MTE 11 2018 - Let the p.d.f. of a continuous random variable X be given byFind t

Question

Let the p.d.f. of a continuous random variable X be given by
$f(x) = \beta e^{^{-\beta x}},x\geq 0$
Find the maximum likelihood estimator of $\beta$ based on a sample X₁ ,X₂ ,....,X_n .

14 Feb 2018

Answer :

Word Count : 225

To find the Maximum Likelihood Estimator (MLE) of $\beta$ for the given probability density function (pdf): \[ f(x) = \beta e^{-\beta x}, \quad x \geq 0 \] we proceed step by step. ### Step 1: Likelihood function For a sample of size $n$ from this distribution, the likelihood function $L(\beta)$ is the joint probability density of the observations $X_1, X_2, ..., X_n$, which is the product of the individual pdfs: \[ L(\beta) = \prod_{i=1}^{n} f(X_i) = \prod_{i=1}^{n} \left( \beta e^{-\beta X_i}

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