Consider a Random Sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$

question 4

Essay

Consider a random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ from the shifted exponential pdf $f ( x ; \lambda , \theta ) = \left\{ \begin{array} { l l } \lambda e ^ { - \lambda ( x - \theta ) } & x \geq \theta \\0 & \text { otherwise }\end{array} \right.$
a. Obtain the maximum likelihood estimators of $\theta \text { and } \lambda$
b. A random sample of size $n = 10$
results in the values 3.12, .65, 2.56, 2.21, 5.45, 3.43, 10.40, 8.94, 17.83, and 1.31, calculate the estimates of $\theta \text { and } \lambda$

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Definitions:

Process

A series of actions or steps taken to achieve a particular end.

Chance Variation

Chance Variation refers to the unpredictability in data or outcomes that arises from random processes inherent in statistical experiments.

Randomly Selected

The process of choosing individuals or items in such a way that each member of a population has an equal chance of being included in the sample.

Normal Fluctuations

Variations in data or statistical measurements that are considered normal or typical within a specific context.