Final answer:
The uniform random variable X is equally distributed over the interval [-1,1]. Y is defined as Y = X² + ε, where ε is a Gaussian distributed variable independent of X. To find the optimum linear estimator for Y, one would normally calculate the covariance and variance of X and use these to derive the coefficients â and ʘ.
Step-by-step explanation:
Let's define the uniform random variable X and the dependent variable Y to answer this student's question. The uniform random variable X is distributed evenly over the interval [-1,1], which means any value within this interval is equally likely to occur. Because of this uniform distribution, we can calculate the mean (μ) as (a+b)/2 and the standard deviation (σ) as (b-a)/√12. For X uniformly distributed over [-1,1], the mean is 0 and the standard deviation is 1/√12.
The variable Y is defined by the equation Y = X² + ε, where ε follows a Gaussian distribution with mean 1 and standard deviation 1 (X∼N(1,1)), and is independent of X. To find the optimum linear mean-squared estimator of Y in terms of aX+b, we look for coefficients â (a-hat) and ʘ (b-hat) that minimize the expected square of the difference between Y and the estimator aX+b.
The process to find these coefficients normally involves calculating the covariance and the variance of X, and using those to derive the estimators. Unfortunately, without a complete set of formulas and steps, we cannot provide the final values for â and ʘ. Nevertheless, this demonstrates the general methodology used to find an optimal linear estimator for a given function of random variables.