Let X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given...

Let X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by FU (x) := P(U ≤ x) =    0 if x ≤ 0 x if 0 < x="">< 1="" 1="" if="" x="" ≥="" 1="" 1. ="" compute="" the="" cdf="" of="" the="" random="" variable="" x1.="" 2. ="" compute="" e(x1)="" and="" v="" ar(x1).="" 3. ="" give="" the="" method="" of="" moments="" estimators="" of="" the="" unknown="" parameters="" a="" and="" b.="" explain="" how="" you="" construct="" these="">

Extracted text: Let X.., Xn be a random sample of size n from an infinite population and assume X1 a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by 0 if x <0 fu(x)="" :="P(U">< x)="x" if="" 0="">< x=""><1 1="" if="" x="">1 Compute the cdf of the random variable X1. Compute E(X1) and Var(X1). Give the method of moments estimators of the unknown parameters a and b. Explain how you construct these estimators! Solution