This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
The probability mass function (PMF) for the random variable X is given as: P(X=0) = (1)/(3) P(X=1) = (1)/(8) P(X=2) = (1)/(2) Let's verify the sum of probabilities: (1)/(3) + (1)/(8) + (1)/(2) = (8)/(24) + (3)/(24) + (12)/(24) = (23)/(24). The probabilities do not sum to 1. We will proceed with the given values. I) Find E[X] and Var[X] Step 1: Calculate the expected value E[X]. The expected value of a discrete random variable is E[X] = x · P(X=x). E[X] = (0 · P(X=0)) + (1 · P(X=1)) + (2 · P(X=2)) E[X] = (0 · (1)/(3)) + (1 · (1)/(8)) + (2 · (1)/(2)) E[X] = 0 + (1)/(8) + 1 E[X] = (1)/(8) + (8)/(8) E[X] = (9)/(8) Step 2: Calculate E[X^2]. The expected value of X^2 is E[X^2] = x^2 · P(X=x). E[X^2] = (0^2 · P(X=0)) + (1^2 · P(X=1)) + (2^2 · P(X=2)) E[X^2] = (0 · (1)/(3)) + (1 · (1)/(8)) + (4 · (1)/(2)) E[X^2] = 0 + (1)/(8) + 2 E[X^2] = (1)/(8) + (16)/(8) E[X^2] = (17)/(8) Step 3: Calculate the variance Var[X]. The variance is given by Var[X] = E[X^2] - (E[X])^2. Var[X] = (17)/(8) - ((9)/(8))^2 Var[X] = (17)/(8) - (81)/(64) To subtract, find a common denominator, which is 64. Var[X] = (17 · 8)/(8 · 8) - (81)/(64) Var[X] = (136)/(64) - (81)/(64) Var[X] = (136 - 81)/(64) Var[X] = (55)/(64) For part I: E[X] = (9)/(8) Var[X] = (55)/(64) II) Find Var[5X - 2] Step 4: Use the property of variance Var[aX + b] = a^2 Var[X]. Here, a=5 and b=-2. Var[5X - 2] = 5^2 · Var[X] Var[5X - 2] = 25 · (55)/(64) Var[5X - 2] = (25 · 55)/(64) Var[5X - 2] = (1375)/(64) For part II: Var[5X - 2] = (1375)/(64) III) Find Var[7X + 47] Step 5: Use the property of variance Var[aX + b] = a^2 Var[X]. Here, a=7 and b=47. Var[7X + 47] = 7^2 · Var[X] Var[7X + 47] = 49 · (55)/(64) Var[7X + 47] = (49 · 55)/(64) Var[7X + 47] = (2695)/(64) For part III: Var[7X + 47] = (2695)/(64) That's 2 down. 3 left today — send the next one.