Which Of The Following Statements Is True

Which Of The Following Statements Is True

Question 1 of 15: 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)2Var(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.

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The question prompts an understanding of basic properties of expectation and variance in probability theory, particularly how they are affected by linear transformations of a random variable. Among the options provided, the true statement is that adding a constant c to each possible value of a discrete random variable X results in a shift in the expected value but does not affect the variance.

Option A is false because variance, which measures the spread or dispersion of a random variable, is unaffected by adding a constant. Variance is invariant under shifts; it depends on how values deviate from the mean, and adding a constant shifts the entire distribution without changing the spread.

Option B is false because the linearity of expectation states that E(aX + b) = aE(X) + b, not (a + b)E(X). The expectation scales with a, and shifts by b; they are additive and multiplicative respectively, not combined into a single factor.

Option C is false because the variance of aX + b equals a2Var(X), as the addition of a constant b has no effect on variance. The expression (a + b)2Var(X) mistakenly combines the shift and scale into a squared term, which is incorrect.

Option D correctly states that when c is added to X, the expected value shifts by c, while the variance remains unchanged. This aligns with fundamental properties of expectation and variance, which are essential concepts in probability and statistics.

Therefore, the statement that adding a constant c to a discrete random variable shifts the expected value by that same amount without changing the variance is accurate.

References

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