IGNOU MST 4 2026 - 6. Let be a random sample from the Exponential distribution

Question

6. Let $X_1, X_2, \dots, X_n$ be a random sample from the Exponential distribution with pdf

$$f(x; \theta) = \theta e^{-\theta x}, x > 0, \theta > 0$

(i) Obtain the maximum likelihood estimator (MLE) of $\theta$ .

(ii) Hence, find the maximum likelihood estimate of $\theta$ for the observed sample:

2.1, 1.6, 3.4, 0.9, 2.5

19 Jan 2026

Answer :

Word Count : 160

The probability density function of the exponential distribution is [ f(x; \theta) = \theta e^{-\theta x}, \quad x>0, \theta>0. ] For a random sample (X_1, X_2, \dots, X_n), the likelihood function is [ L(\theta) = \prod_{i=1}^{n} f(X_i;\theta) = \prod_{i=1}^{n} \theta e^{-\theta X_i} = \theta^n e^{-\theta \sum_{i=1}^{n} X_i}. ] The log-likelihood function is [ \ell(\theta) = \ln L(\theta) = n

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