IGNOU MST 4 2018 - If the magnitude of the earthquakes recorded in a region of a country

Question

If the magnitude of the earthquakes recorded in a region of a country follows a distribution with parameter θ

$\frac{1}{\theta }e^\frac{-x}{\theta },x\geq 0,\theta \geq 0$ then find

i) maximum likelihood estimator of the parameter θ,

ii) maximum likelihood estimate of the parameter on the basis of the following data:

Magnitude of the Earthquakes

(on the Richter scale)

6.7

7.7

5.6

7.3

6.7

6.6

7.8

6.7

6.2

5.2

iii) Show that maximum likelihood estimator is unbiased and sufficient for parameter θ.

16 Feb 2018

Answer :

Word Count : 489

We are asked to manually find the Maximum Likelihood Estimator (MLE) for the parameter $\theta$ of an exponential distribution, then compute it for given earthquake magnitudes, and show its properties. Let's solve step by step. --- ### Given PDF of the exponential distribution: $$ f(x; \theta) = \frac{1}{\theta} e^{-x/\theta}, \quad x \ge 0, \theta > 0 $$ Data (magnitudes of earthquakes): $$ 6.7, 7.7, 5.6, 7.3, 6.7, 6.6, 7.8, 6.7, 6.2, 5.2 $$ --- ## Step i: Maximum Likelihood Estimator (MLE) of $\theta$ 1. Likelihood function For a sample $x_1, x_2, \dots, x_n$, the likelihood is: $$ L(\theta) = \prod_{i=1}^n f(x_i;\theta) = \prod_{i=1}^n \frac{1}{\theta} e^{-x_i/\theta} = \frac{1}{\theta^n} e^{-\sum x_i / \theta} $$

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S. No.	Breaking Pressure
S. No.	Iron	Copper
1	72.7	73
2	69.6	67.2
3	83.4	75.3
4	78.9	61.4
5	75	74
6	71.6	69.5
7	85.7	69.8
8	73.5	73.8
9	70.4	68
10	84.2	76.1

Educational attainment	Income Categories			Total
Educational attainment	Low	Middle	High	Total
No college UG PG Doctoral	32 13 43 12	20 16 51 13	23 1 60 15	75 30 154 41
Total	100	100	100	300