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.

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.