Let X be a discrete random variable. The probability distribution of X is given below ( array |l|c|c|c| X & 30 & 10 & -10 P ( X ) & 1 5 & ~A & ~B array ) and E ( X )=4 , then the value of AB is equal to

Step-by-step solution

The probability distribution is defined with A + B = 4 5 from the total probability constraint. Given E(X) = 4 , apply the expectation formula: 4 = 30 1 5 + 10A - 10B , simplifying to 5A - 5B = -1 . Multiply the total probability equation by 5: 5A + 5B = 4 . Adding this to 5A - 5B = -1 yields 10A = 3 , so A = 3 10 . Substitute into A + B = 4 5 : B = 4 5 - 3 10 = 1 2 . Then AB = 3 10 1 2 = 3 20 , matching option D .