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StatisticsQ&A Library1. Define a random variable X as the proportion of time that a student spends on a certain test. Research suggests that the pdf of x is f(x) = (θ + 1)xθ , 0 ≤ x ≤ 1. For the parameter θ, we know θ > −1, but its exact value is still unknown. To estimate the parameter, a random sample of 10 students is obtained, and their proportions of time are: 0.92, 0.79, 0.90, 0.65, 0.86, 0.47, 0.73, 0.97, 0.94, 0.77. Find the Maximum Likelihood Estimator (MLE) of θ.Question

Asked Nov 13, 2019

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1. Define a random variable X as the proportion of time that a student spends on a certain test. Research suggests that the pdf of x is f(x) = (θ + 1)xθ , 0 ≤ x ≤ 1. For the parameter θ, we know θ > −1, but its exact value is still unknown. To estimate the parameter, a random sample of 10 students is obtained, and their proportions of time are: 0.92, 0.79, 0.90, 0.65, 0.86, 0.47, 0.73, 0.97, 0.94, 0.77. Find the Maximum Likelihood Estimator (MLE) of θ.

Step 1

**Likelihood function:**

The pdf is considered as f(x)=(*θ+1*)*x*^{θ}*.*

The likelihood function of the distribution is:

Step 2

**Log likelihood function:**

Thus, the log likelihood function obtained by taking logarithm of the likelihood function is:

Step 3

** ****Differentiating the log likelihood function with respect to θ:**

The maximum likelihood estimator of *θ* is obtained by differentiating the log ...

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