Let X1, X2, ...,Xn be a random sample of size n from a distribution with probability density function f(x; 8) = } 0(1 + x)-(1+0), x >0 elsewhere Find the Maximum likelihood estimator (MLE) of 0.

What are the expected value and the standard deviation of the area of the circle whose radius is a random variable X with density function f(x) = 1 for 0 < x < 1 and f(x) = 0 otherwise ?

Estimate the unknown parameter from a sample 3, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7 drawn from a discrete distribution with the probability mass functiontion 0 (P(3) P(7) = 1-0 0 - Compute the estimator of 0 use the method of moments.

Let X be a random variable with a uniform distribution over the interval [-2 , 4]. Determine the mean of X. The answer should be a number rounded to five decimal places