Question

Show all work. Answers are given.Use double integrals to answer questions

random variables A and B have joint probability density function given by

fA,B (a, b) = ab + a + b + 1, −1 ≤ a ≤ 1, −1 ≤ b ≤ 0

Use double integrals to answer the questions
(a) P(A ≤ ½).

Answer is 9/16

(b) P(−½ ≤ A ≤ ½ , −½ ≤ B ≤ ½ )

Answer is ⅜

(c) P(A ≥ B + 1)

Answer is 7/24

(d) calculate marginals fA(a) and fB(b).

Answer is fA(a) = (½)a + ½ if -1 ≤ a ≤ 1, fB(b) = 2b + 2 if -1 ≤ b ≤ 0

(e) Using marginal fa(a) to find P(A ≤ ½).

Answer is 9/16

(f) Show that A and B are independent by checking if fA,B = fAfB.

(g) use factoring arguments to show that A and B are independent.

Answer 1

any query, please ask