![Given Y = 3 - 2X, E[X] = 5, and Var(X) = 4, which statement is correct?](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1777400335010-78b123038791ac8e.png&w=3840&q=75)
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Calculate the expected value of Y, E[Y]. Given the linear transformation Y = aX + b, the expected value is E[Y] = aE[X] + b. In this case, Y = 3 - 2X, so a = -2 and b = 3. We are given E[X] = 5. E[Y] = E[3 - 2X] = 3 - 2E[X] E[Y] = 3 - 2(5) E[Y] = 3 - 10 E[Y] = -7 Step 2: Calculate the variance of Y, Var(Y). Given the linear transformation Y = aX + b, the variance is Var(Y) = a^2 Var(X). In this case, Y = 3 - 2X, so a = -2. We are given Var(X) = 4. Var(Y) = Var(3 - 2X) = (-2)^2 Var(X) Var(Y) = (4)(4) Var(Y) = 16 Step 3: Compare the calculated values with the given options. Our calculations show E[Y] = -7 and Var(Y) = 16. Option D states: E[Y] = 3 - 2E[X] = -7 and Var(Y) = (-2)^2 Var(X) = 16. This matches our results. The correct statement is D. D. E[Y] = 3 - 2E[X] = -7 and Var(Y) = (-2)^2 Var(X) = 16 That's 2 down. 3 left today — send the next one.