Let X1, . . . , Xn be independent random variables with uniform distribution U[0, 1].(a) Determine α that satisfies the limit in probability limn→+∞X21 + X22 + · · · + X2nn = α.[6 marks](b) Derive the probability density function (pdf) of Yi = √Xi, with i = 1, .., n. [5 marks](c) Find β that satisfies the limit in probability limn→+∞√X1 + √X2 + · · · + √Xnn = β

(a) The random variables X1, X2, ..., Xn are independent and uniformly distributed on [0, 1]. The expected value of X^2 for a uniform distribution on [0, 1] is given by the integral from 0 to 1 of x^2 dx, which equals 1/3. Therefore, the expected value of the sum of the squares of these random variables is n/3. By the law of large numbers, as n approaches infinity, the average of these random variables approaches their expected value. Therefore, α = 1/3.

(b) The random variable Yi = √Xi has a probability density function (pdf) that can be derived from the pdf of Xi. The pdf of Xi is f(x) = 1 for 0 <= x <= 1 and 0 otherwise. The transformation y = √x gives x = y^2. The Jacobian of this transformation is dx/dy = 2y. Therefore, the pdf of Yi is g(y) = f(y^2) * |dx/dy| = 2y for 0 <= y <= 1 and 0 otherwise.

(c) The random variables √X1, √X2, ..., √Xn are independent and have the same pdf as derived in part (b). The expected value of √X for a uniform distribution on [0, 1] is given by the integral from 0 to 1 of 2y^2 dy, which equals 2/3. Therefore, the expected value of the sum of the square roots of these random variables is 2n/3. By the law of large numbers, as n approaches infinity, the average of these random variables approaches their expected value. Therefore, β = 2/3.