Question

Suppose the random variable $X$ has the density function $$ f(x)= \begin{cases}(1+\lambda) x^\lambda, & 0<x<1 \\ 0 & \text { otherwise. }\end{cases} $$ Show that the maximum likelihood estimate of $\lambda$ based on a given random sample of size $n$ is given by $$ \hat{\lambda}=-\left(1+\frac{n}{\sum_{i=1}^n \ln x_i}\right) . $$

   Suppose the random variable $X$ has the density function
$$
f(x)= \begin{cases}(1+\lambda) x^\lambda, & 0<x<1 \\ 0 & \text { otherwise. }\end{cases}
$$
Show that the maximum likelihood estimate of $\lambda$ based on a given random sample of size $n$ is given by
$$
\hat{\lambda}=-\left(1+\frac{n}{\sum_{i=1}^n \ln x_i}\right) .
$$

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Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 7, Problem 1

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Step 1

Given the density function \( f(x) = (1+\lambda) x^\lambda \) for \( 0 < x < 1 \) and \( f(x) = 0 \) otherwise, the likelihood function \( L(\lambda) \) for a sample \( x_1, x_2, \ldots, x_n \) is the product of the density function evaluated at each sample Show more…

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Suppose the random variable $X$ has the density function $$ f(x)= \begin{cases}(1+\lambda) x^\lambda, & 0<x<1 \\ 0 & \text { otherwise. }\end{cases} $$ Show that the maximum likelihood estimate of $\lambda$ based on a given random sample of size $n$ is given by $$ \hat{\lambda}=-\left(1+\frac{n}{\sum_{i=1}^n \ln x_i}\right) . $$

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