Question

Let $X_1, X_2, \dots, X_n$ is a random sample from a distribution with density function $f(x; \beta) = \begin{cases} \frac{x^6 e^{-\frac{x}{\beta}}}{\Gamma(7)\beta^7} & \text{for } 0 < x < \infty \\ 0 & \text{otherwise} \end{cases}$ then what is the maximum likelihood estimator of $\beta$? a. 7X b. X/7 c. 6X d. $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$

          Let $X_1, X_2, \dots, X_n$ is a random sample from a distribution with density function
$f(x; \beta) = \begin{cases} \frac{x^6 e^{-\frac{x}{\beta}}}{\Gamma(7)\beta^7} & \text{for } 0 < x < \infty \\ 0 & \text{otherwise} \end{cases}$
then what is the maximum likelihood estimator of $\beta$?
a. 7X
b. X/7
c. 6X
d. $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$

Let X1, X2, …, Xn is a random sample from a distribution with density function
f(x; β) = (x^6 e^-(x)/(β))/(Γ(7)β^7) for 0 < x < ∞
0 otherwise
then what is the maximum likelihood estimator of β?
a. 7X
b. X/7
c. 6X
d. X̅ = (1)/(n)∑i=1^n Xi

Added by Emily B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

08/10/2023

Step 1

Step 1: The likelihood function is given by: $$L(\beta) = \prod_{i=1}^n f(x_i; \beta) = \prod_{i=1}^n \frac{x_i^6 e^{-\frac{x_i}{\beta}}}{\Gamma(7)\beta^7} = \frac{1}{(\Gamma(7)\beta^7)^n} \prod_{i=1}^n x_i^6 e^{-\frac{x_i}{\beta}}$$ Show more…

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Let $X_1, X_2, ..., X_n$ is a random sample from a distribution with density function $$f(x; \beta) = \begin{cases} \frac{x^6 e^{-\frac{x}{\beta}}}{\Gamma(7)\beta^7} & \text{for } 0 < x < \infty \\ 0 & \text{otherwise} \end{cases}$$ then what is the maximum likelihood estimator of $\beta$? a. 7X b. X/7 c. 6X d. $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$