1. (10 points) Consider the polynomial p(x) = x+Pr+Q, where P and Q are independent random...

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1. (10 points) Consider the polynomial p(x) = x+Pr+Q, where P and Q are independent random variables with P~U(0, 1) and Q~U(0, 1). a) What is the probability that p(x) has two real roots? b) Let the random variable R denote the minimal value which is attained by the polynomial p(x), i.e. R= min{x+Pr+Q: xER}. Determine E(R). (Hint: you could differentiate to find the minimum of p(x).) c) Let the random variable N denote the number of real roots of p(x). What is E(N)? 1. (10 points) Consider the polynomial p(x) = x+Pr+Q, where P and Q are independent random variables with P~U(0, 1) and Q~U(0, 1). a) What is the probability that p(x) has two real roots? b) Let the random variable R denote the minimal value which is attained by the polynomial p(x), i.e. R= min{x+Pr+Q: xER}. Determine E(R). (Hint: you could differentiate to find the minimum of p(x).) c) Let the random variable N denote the number of real roots of p(x). What is E(N)?