Which Of The Following Statements Is True 797863

which Of The Following Statements Is True

Question 1 of 10 150 Points which Of The Following Statements Is True

Question 1 of 10 150 Points which Of The Following Statements Is True

Question 1 of 10 150 Points Which of the following statements is true? A. If a constant c is added to each possible value of a discrete random variable X, then the variance of X will be shifted by that same constant amount. B. For any discrete random variable X and constants a and b, E(aX + b) = (a + b)E(X). C. For any discrete random variable X and constants a and b, Var(aX + b) = (a + b)^2 Var(X). D. If a constant c is added to each possible value of a discrete random variable X, then the expected value of X will be shifted by that same constant amount.

Paper For Above instruction

The question explores fundamental properties of random variables, particularly focusing on the effects of constants on variance and expectation. Understanding these properties is crucial for statistical analysis and probabilistic modeling. The core concepts involve how linear transformations impact the mean and variance of a random variable. The options given test knowledge about the invariance or change in these statistical measures when constant shifts or linear transformations occur.

Option A suggests that adding a constant c to a discrete random variable X shifts its variance by the same amount. However, this is incorrect because variance measures dispersion and is unaffected by shifts; adding a constant shifts the mean but does not alter variability. Option B claims that the expectation of a linear transformation E(aX + b) equals (a + b)E(X), which is false; the correct formula is E(aX + b) = aE(X) + b. Option C states that the variance scales as (a + b)^2 Var(X), but variance is unaffected by adding a constant and scales only with the coefficient a, not the additive b. Lastly, option D correctly states that adding a constant c shifts the expected value by c, aligning with the linearity of expectation, which confirms E(X + c) = E(X) + c.

Therefore, the true statement among the options given is option D, which accurately describes how expected value behaves under the addition of a constant. Understanding this property is critical when modeling random variables, as it simplifies analysis involving shifts without affecting variability.

Answer:

The correct answer is D: If a constant c is added to each possible value of a discrete random variable X, then the expected value of X will be shifted by that same constant amount.

References

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